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The volume of a sphere is $36\pi$ cubic units. What is the radius of the sphere, in units? | nvidia_math | [
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] | 0.920815 | 0.879924 | 0.040891 | <think>
Alright, so I have this problem here: The volume of a sphere is 36π cubic units. I need to find the radius of the sphere. Hmm, okay. I think I remember the formula for the volume of a sphere. Let me try to recall. Is it (4/3)πr³? Yeah, that sounds right. So the volume V is equal to (4/3)π times the radius cubed.
Alright, so they've given me the volume, which is 36π. So I can set up the equation as 36π = (4/3)πr³. Now, I need to solve for r. Let me think about how to do that.
First, I can probably divide both sides by π to simplify. That makes sense because π is on both sides, right? So dividing both sides by π would cancel it out. Let me write that down:
36π / π = (4/3)πr³ / π
Simplifying both sides, the π cancels out on both sides, so we get:
36 = (4/3)r³
Okay, so now the equation is just 36 equals (4/3)r³. Now I need to solve for r³. To do that, I can multiply both sides by the reciprocal of 4/3, which is 3/4. Let me see:
36 * (3/4) = (4/3)r³ * (3/4)
On the left side, multiplying 36 by 3/4. Hmm, let's compute that. 36 divided by 4 is 9, then 9 times 3 is 27. So, 36*(3/4) = 27.
On the right side, the (4/3) and (3/4) multiply together to give 1, so we're just left with r³. So:
27 = r³
Okay, so now we have r³ = 27. To find r, I need to take the cube root of both sides. The cube root of 27 is 3, because 3*3*3 is 27.
Therefore, r = 3.
Wait, let me check my steps again to make sure I didn't make a mistake. I started with the formula for the volume, plugged in the given volume, divided both sides by π correctly, yeah that gives 36 = (4/3)r³. Then I multiplied 36 by 3/4 to isolate r³, got 27, so r is 3. Seems right.
Hmm, another way to do it: Instead of dividing by π first, I could have kept it and then canceled it out when multiplying both sides by π. Let me try that approach to double-check.
Starting again with V = (4/3)πr³. So 36π = (4/3)πr³. If I multiply both sides by 3/(4π), that would isolate r³. Let's calculate:
36π * (3/(4π)) = (4/3)πr³ * (3/(4π))
Simplify left side: 36π*(3)/(4π). The π cancels, 36 and 4 can be simplified. 36 divided by 4 is 9, 9*3 is 27, so 27.
Right side: (4/3)*(3/4) is 1; π cancels with π, so right side is r³.
So again, 27 = r³, so r=3.
Okay, same answer. Good.
I think that's solid then. So the radius is 3 units.
**Final Answer**
The radius of the sphere is \boxed{3} units.
</think>
The volume of a sphere is given as \(36\pi\) cubic units. To find the radius, we use the formula for the volume of a sphere, which is \(V = \frac{4}{3}\pi r^3\).
Starting with the given volume:
\[ 36\pi = \frac{4}{3}\pi r^3 \]
We divide both sides by \(\pi\):
\[ 36 = \frac{4}{3}r^3 \]
Next, we multiply both sides by \(\frac{3}{4}\) to isolate \(r^3\):
\[ 36 \times \frac{3}{4} = r^3 \]
\[ 27 = r^3 \]
Taking the cube root of both sides, we find:
\[ r = \sqrt[3]{27} = 3 \]
Thus, the radius of the sphere is \(\boxed{3}\) units. | 1 | [
{
"from": "human",
"value": "The volume of a sphere is $36\\pi$ cubic units. What is the radius of the sphere, in units?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I have this problem here: The volume of a sphere is 36π cubic units. I need to find the radius of the sphere. Hmm, okay. I think I remember the formula for the volume of a sphere. Let me try to recall. Is it (4/3)πr³? Yeah, that sounds right. So the volume V is equal to (4/3)π times the radius cubed. \n\nAlright, so they've given me the volume, which is 36π. So I can set up the equation as 36π = (4/3)πr³. Now, I need to solve for r. Let me think about how to do that. \n\nFirst, I can probably divide both sides by π to simplify. That makes sense because π is on both sides, right? So dividing both sides by π would cancel it out. Let me write that down:\n\n36π / π = (4/3)πr³ / π\n\nSimplifying both sides, the π cancels out on both sides, so we get:\n\n36 = (4/3)r³\n\nOkay, so now the equation is just 36 equals (4/3)r³. Now I need to solve for r³. To do that, I can multiply both sides by the reciprocal of 4/3, which is 3/4. Let me see:\n\n36 * (3/4) = (4/3)r³ * (3/4)\n\nOn the left side, multiplying 36 by 3/4. Hmm, let's compute that. 36 divided by 4 is 9, then 9 times 3 is 27. So, 36*(3/4) = 27. \n\nOn the right side, the (4/3) and (3/4) multiply together to give 1, so we're just left with r³. So:\n\n27 = r³\n\nOkay, so now we have r³ = 27. To find r, I need to take the cube root of both sides. The cube root of 27 is 3, because 3*3*3 is 27.\n\nTherefore, r = 3. \n\nWait, let me check my steps again to make sure I didn't make a mistake. I started with the formula for the volume, plugged in the given volume, divided both sides by π correctly, yeah that gives 36 = (4/3)r³. Then I multiplied 36 by 3/4 to isolate r³, got 27, so r is 3. Seems right.\n\nHmm, another way to do it: Instead of dividing by π first, I could have kept it and then canceled it out when multiplying both sides by π. Let me try that approach to double-check.\n\nStarting again with V = (4/3)πr³. So 36π = (4/3)πr³. If I multiply both sides by 3/(4π), that would isolate r³. Let's calculate:\n\n36π * (3/(4π)) = (4/3)πr³ * (3/(4π))\n\nSimplify left side: 36π*(3)/(4π). The π cancels, 36 and 4 can be simplified. 36 divided by 4 is 9, 9*3 is 27, so 27.\n\nRight side: (4/3)*(3/4) is 1; π cancels with π, so right side is r³.\n\nSo again, 27 = r³, so r=3.\n\nOkay, same answer. Good.\n\nI think that's solid then. So the radius is 3 units.\n\n**Final Answer**\nThe radius of the sphere is \\boxed{3} units.\n</think>\n\nThe volume of a sphere is given as \\(36\\pi\\) cubic units. To find the radius, we use the formula for the volume of a sphere, which is \\(V = \\frac{4}{3}\\pi r^3\\).\n\nStarting with the given volume:\n\\[ 36\\pi = \\frac{4}{3}\\pi r^3 \\]\n\nWe divide both sides by \\(\\pi\\):\n\\[ 36 = \\frac{4}{3}r^3 \\]\n\nNext, we multiply both sides by \\(\\frac{3}{4}\\) to isolate \\(r^3\\):\n\\[ 36 \\times \\frac{3}{4} = r^3 \\]\n\\[ 27 = r^3 \\]\n\nTaking the cube root of both sides, we find:\n\\[ r = \\sqrt[3]{27} = 3 \\]\n\nThus, the radius of the sphere is \\(\\boxed{3}\\) units."
}
] |
A circular spinner for a game has a radius of 10 cm. The probability of winning on one spin of this spinner is $\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\pi$.
[asy]import graph;
draw(Circle((0,0),25),black);
draw((0,0)--(7,18),Arrow);
draw((0,0)--(0,25));
draw((0,0)--(15,-20));
label("WIN",(10,10),S);
label("LOSE",(-8,-8),N);
dot((0,0));
[/asy] | tiger_math | [
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] | 0.920414 | 0.876023 | 0.044391 | <think>
First, I recognize that the spinner is circular and has a radius of 10 cm. To find the area of the entire spinner, I'll use the formula for the area of a circle, which is π multiplied by the radius squared. So, the total area is π times 10 cm squared, which equals 100π square centimeters.
Next, the probability of winning on one spin is given as 2/5. Probability is the ratio of the favorable area (the WIN sector) to the total area of the spinner. Let's denote the area of the WIN sector as A. Therefore, I can set up the equation A divided by 100π equals 2/5.
To solve for A, I'll multiply both sides of the equation by 100π. This gives me A equals (2/5) multiplied by 100π. Calculating that, (2/5) times 100 is 40, so the area of the WIN sector is 40π square centimeters.
</think>
To determine the area of the **WIN** sector on the spinner, follow these steps:
1. **Calculate the Total Area of the Spinner:**
The spinner is a circle with a radius of 10 cm. The formula for the area of a circle is:
\[
\text{Area} = \pi r^2
\]
Plugging in the given radius:
\[
\text{Total Area} = \pi (10\, \text{cm})^2 = 100\pi\, \text{sq cm}
\]
2. **Apply the Probability to Find the WIN Sector Area:**
The probability of landing on the **WIN** sector is \(\frac{2}{5}\). Probability is equal to the ratio of the favorable area (WIN sector) to the total area. Let \(A\) be the area of the WIN sector. Thus:
\[
\frac{A}{100\pi} = \frac{2}{5}
\]
3. **Solve for \(A\):**
\[
A = \frac{2}{5} \times 100\pi = 40\pi\, \text{sq cm}
\]
**Final Answer:**
\[
\boxed{40\pi}
\] | 6 | [
{
"from": "human",
"value": "A circular spinner for a game has a radius of 10 cm. The probability of winning on one spin of this spinner is $\\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\\pi$.\n\n[asy]import graph;\n\ndraw(Circle((0,0),25),black);\n\ndraw((0,0)--(7,18),Arrow);\n\ndraw((0,0)--(0,25));\n\ndraw((0,0)--(15,-20));\n\nlabel(\"WIN\",(10,10),S);\n\nlabel(\"LOSE\",(-8,-8),N);\n\ndot((0,0));\n\n[/asy]"
},
{
"from": "gpt",
"value": "<think>\nFirst, I recognize that the spinner is circular and has a radius of 10 cm. To find the area of the entire spinner, I'll use the formula for the area of a circle, which is π multiplied by the radius squared. So, the total area is π times 10 cm squared, which equals 100π square centimeters.\n\nNext, the probability of winning on one spin is given as 2/5. Probability is the ratio of the favorable area (the WIN sector) to the total area of the spinner. Let's denote the area of the WIN sector as A. Therefore, I can set up the equation A divided by 100π equals 2/5.\n\nTo solve for A, I'll multiply both sides of the equation by 100π. This gives me A equals (2/5) multiplied by 100π. Calculating that, (2/5) times 100 is 40, so the area of the WIN sector is 40π square centimeters.\n</think>\n\nTo determine the area of the **WIN** sector on the spinner, follow these steps:\n\n1. **Calculate the Total Area of the Spinner:**\n \n The spinner is a circle with a radius of 10 cm. The formula for the area of a circle is:\n \\[\n \\text{Area} = \\pi r^2\n \\]\n Plugging in the given radius:\n \\[\n \\text{Total Area} = \\pi (10\\, \\text{cm})^2 = 100\\pi\\, \\text{sq cm}\n \\]\n\n2. **Apply the Probability to Find the WIN Sector Area:**\n \n The probability of landing on the **WIN** sector is \\(\\frac{2}{5}\\). Probability is equal to the ratio of the favorable area (WIN sector) to the total area. Let \\(A\\) be the area of the WIN sector. Thus:\n \\[\n \\frac{A}{100\\pi} = \\frac{2}{5}\n \\]\n \n3. **Solve for \\(A\\):**\n \\[\n A = \\frac{2}{5} \\times 100\\pi = 40\\pi\\, \\text{sq cm}\n \\]\n \n**Final Answer:**\n\\[\n\\boxed{40\\pi}\n\\]"
}
] |
A cylindrical container with a diameter of 8 inches and a height of 15 inches needs to be wrapped in a rectangular sheet of cardboard for shipping. The cardboard sheet should be large enough to completely cover the curved surface of the cylinder, and the height of the sheet should match the height of the cylinder. If the cardboard sheet is sold in integer dimensions only, what is the minimum area of the sheet needed to wrap the cylinder, in square inches? | nvidia_math | [
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] | 0.914088 | 0.873598 | 0.04049 | <think>
First, I need to determine the dimensions of the rectangular cardboard sheet required to wrap the cylindrical container.
The height of the cylinder is 15 inches, and this will be the height of the cardboard sheet.
Next, I'll calculate the circumference of the cylinder using its diameter. The circumference is π times the diameter, which is π * 8 inches, approximately 25.13 inches. Since the cardboard can only be purchased in integer dimensions, I'll round this up to the next whole number, which is 26 inches.
Now, the area of the cardboard sheet is the product of its height and width. Multiplying 15 inches by 26 inches gives 390 square inches.
Therefore, the minimum area of the sheet needed to wrap the cylinder is 390 square inches.
</think>
To determine the minimum area of the rectangular cardboard sheet needed to wrap the cylindrical container, follow these steps:
### **1. Understand the Dimensions of the Cylinder**
- **Diameter (D):** 8 inches
- **Height (H):** 15 inches
### **2. Calculate the Circumference of the Cylinder**
The **circumference (C)** of the cylinder is given by the formula:
\[
C = \pi \times D
\]
Plugging in the values:
\[
C = \pi \times 8 \approx 3.1416 \times 8 \approx 25.1328 \text{ inches}
\]
Since the cardboard's width must cover the entire circumference, and the width must be an integer, we round up to the nearest whole number:
\[
\text{Width of Cardboard} = 26 \text{ inches}
\]
### **3. Determine the Height of the Cardboard**
The height of the cardboard should match the height of the cylinder:
\[
\text{Height of Cardboard} = 15 \text{ inches}
\]
### **4. Calculate the Area of the Cardboard**
The area (**A**) of the rectangular cardboard is:
\[
A = \text{Width} \times \text{Height} = 26 \times 15 = 390 \text{ square inches}
\]
### **Final Answer**
The minimum area of the cardboard sheet needed is:
\[
\boxed{390}
\] | 7 | [
{
"from": "human",
"value": "A cylindrical container with a diameter of 8 inches and a height of 15 inches needs to be wrapped in a rectangular sheet of cardboard for shipping. The cardboard sheet should be large enough to completely cover the curved surface of the cylinder, and the height of the sheet should match the height of the cylinder. If the cardboard sheet is sold in integer dimensions only, what is the minimum area of the sheet needed to wrap the cylinder, in square inches?"
},
{
"from": "gpt",
"value": "<think>\nFirst, I need to determine the dimensions of the rectangular cardboard sheet required to wrap the cylindrical container.\n\nThe height of the cylinder is 15 inches, and this will be the height of the cardboard sheet.\n\nNext, I'll calculate the circumference of the cylinder using its diameter. The circumference is π times the diameter, which is π * 8 inches, approximately 25.13 inches. Since the cardboard can only be purchased in integer dimensions, I'll round this up to the next whole number, which is 26 inches.\n\nNow, the area of the cardboard sheet is the product of its height and width. Multiplying 15 inches by 26 inches gives 390 square inches.\n\nTherefore, the minimum area of the sheet needed to wrap the cylinder is 390 square inches.\n</think>\n\nTo determine the minimum area of the rectangular cardboard sheet needed to wrap the cylindrical container, follow these steps:\n\n### **1. Understand the Dimensions of the Cylinder**\n- **Diameter (D):** 8 inches\n- **Height (H):** 15 inches\n\n### **2. Calculate the Circumference of the Cylinder**\nThe **circumference (C)** of the cylinder is given by the formula:\n\\[\nC = \\pi \\times D\n\\]\nPlugging in the values:\n\\[\nC = \\pi \\times 8 \\approx 3.1416 \\times 8 \\approx 25.1328 \\text{ inches}\n\\]\nSince the cardboard's width must cover the entire circumference, and the width must be an integer, we round up to the nearest whole number:\n\\[\n\\text{Width of Cardboard} = 26 \\text{ inches}\n\\]\n\n### **3. Determine the Height of the Cardboard**\nThe height of the cardboard should match the height of the cylinder:\n\\[\n\\text{Height of Cardboard} = 15 \\text{ inches}\n\\]\n\n### **4. Calculate the Area of the Cardboard**\nThe area (**A**) of the rectangular cardboard is:\n\\[\nA = \\text{Width} \\times \\text{Height} = 26 \\times 15 = 390 \\text{ square inches}\n\\]\n\n### **Final Answer**\nThe minimum area of the cardboard sheet needed is:\n\\[\n\\boxed{390}\n\\]"
}
] |
the length of a rectangle is increased by 25 % and its breadth is decreased by 20 % . what is the effect on its area ? | tiger_math | [
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] | 0.938394 | 0.890463 | 0.047931 | <think>
I need to determine how changing the length and breadth of a rectangle affects its area. Let's denote the original length as L and the original breadth as B. The original area of the rectangle is then A = L × B.
When the length is increased by 25%, the new length becomes 1.25L. Similarly, when the breadth is decreased by 20%, the new breadth becomes 0.8B.
Now, calculating the new area with the updated dimensions: A' = 1.25L × 0.8B. This simplifies to A' = (1.25 × 0.8) × L × B, which is A' = 1.0 × L × B. Therefore, the new area is equal to the original area, indicating no change in the area of the rectangle.
</think>
To determine the effect on the area of the rectangle when its length is increased by 25% and its breadth is decreased by 20%, follow these steps:
1. **Original Area:**
- Let the original length be \( L \) and the original breadth be \( B \).
- Original Area \( A_1 = L \times B \).
2. **New Dimensions:**
- **Length increased by 25%:**
\[
\text{New Length} = L + \left( \frac{25}{100} \times L \right) = 1.25L
\]
- **Breadth decreased by 20%:**
\[
\text{New Breadth} = B - \left( \frac{20}{100} \times B \right) = 0.8B
\]
3. **New Area:**
- Calculate the area with the new dimensions:
\[
A_2 = \text{New Length} \times \text{New Breadth} = 1.25L \times 0.8B
\]
- Simplify the expression:
\[
A_2 = (1.25 \times 0.8) \times L \times B = 1.0 \times L \times B = A_1
\]
4. **Conclusion:**
- Since \( A_2 = A_1 \), there is **no change** in the area of the rectangle.
\[
\boxed{\text{No change in the area}}
\] | 9 | [
{
"from": "human",
"value": "the length of a rectangle is increased by 25 % and its breadth is decreased by 20 % . what is the effect on its area ?"
},
{
"from": "gpt",
"value": "<think>\nI need to determine how changing the length and breadth of a rectangle affects its area. Let's denote the original length as L and the original breadth as B. The original area of the rectangle is then A = L × B.\n\nWhen the length is increased by 25%, the new length becomes 1.25L. Similarly, when the breadth is decreased by 20%, the new breadth becomes 0.8B.\n\nNow, calculating the new area with the updated dimensions: A' = 1.25L × 0.8B. This simplifies to A' = (1.25 × 0.8) × L × B, which is A' = 1.0 × L × B. Therefore, the new area is equal to the original area, indicating no change in the area of the rectangle.\n</think>\n\nTo determine the effect on the area of the rectangle when its length is increased by 25% and its breadth is decreased by 20%, follow these steps:\n\n1. **Original Area:**\n - Let the original length be \\( L \\) and the original breadth be \\( B \\).\n - Original Area \\( A_1 = L \\times B \\).\n\n2. **New Dimensions:**\n - **Length increased by 25%:**\n \\[\n \\text{New Length} = L + \\left( \\frac{25}{100} \\times L \\right) = 1.25L\n \\]\n - **Breadth decreased by 20%:**\n \\[\n \\text{New Breadth} = B - \\left( \\frac{20}{100} \\times B \\right) = 0.8B\n \\]\n\n3. **New Area:**\n - Calculate the area with the new dimensions:\n \\[\n A_2 = \\text{New Length} \\times \\text{New Breadth} = 1.25L \\times 0.8B\n \\]\n - Simplify the expression:\n \\[\n A_2 = (1.25 \\times 0.8) \\times L \\times B = 1.0 \\times L \\times B = A_1\n \\]\n\n4. **Conclusion:**\n - Since \\( A_2 = A_1 \\), there is **no change** in the area of the rectangle.\n\n\\[\n\\boxed{\\text{No change in the area}}\n\\]"
}
] |
A rectangular prism has a volume of 1800 cubic units. The length and width of the prism are equal, and the height is twice the length. What is the length of the prism? | tiger_math | [
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] | 0.924042 | 0.886635 | 0.037407 | <think>
First, I recognize that the volume of a rectangular prism is calculated by multiplying its length, width, and height.
Since the problem states that the length and width are equal, I'll denote the length as \( l \) and the width as \( l \) as well. The height is twice the length, so I'll represent the height as \( 2l \).
Now, I can set up the volume formula with the given information:
\[
\text{Volume} = l \times l \times 2l = 2l^3
\]
Given that the volume is 1800 cubic units, I'll set up the equation:
\[
2l^3 = 1800
\]
To solve for \( l \), I'll divide both sides by 2:
\[
l^3 = 900
\]
Taking the cube root of both sides gives:
\[
l = \sqrt[3]{900} = 10
\]
So, the length of the prism is 10 units.
</think>
To find the length of the rectangular prism, let's follow these steps:
1. **Define the Variables:**
- Let \( l \) be the length of the prism.
- Since the length and width are equal, the width is also \( l \).
- The height is twice the length, so the height is \( 2l \).
2. **Write the Volume Formula:**
The volume \( V \) of a rectangular prism is given by:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
Substituting the given values:
\[
1800 = l \times l \times 2l
\]
Simplifying:
\[
1800 = 2l^3
\]
3. **Solve for \( l \):**
\[
2l^3 = 1800
\]
\[
l^3 = \frac{1800}{2} = 900
\]
\[
l = \sqrt[3]{900}
\]
\[
l = 10
\]
4. **Final Answer:**
The length of the prism is:
\[
\boxed{10}
\] | 10 | [
{
"from": "human",
"value": "A rectangular prism has a volume of 1800 cubic units. The length and width of the prism are equal, and the height is twice the length. What is the length of the prism?"
},
{
"from": "gpt",
"value": "<think>\nFirst, I recognize that the volume of a rectangular prism is calculated by multiplying its length, width, and height.\n\nSince the problem states that the length and width are equal, I'll denote the length as \\( l \\) and the width as \\( l \\) as well. The height is twice the length, so I'll represent the height as \\( 2l \\).\n\nNow, I can set up the volume formula with the given information:\n\\[\n\\text{Volume} = l \\times l \\times 2l = 2l^3\n\\]\nGiven that the volume is 1800 cubic units, I'll set up the equation:\n\\[\n2l^3 = 1800\n\\]\nTo solve for \\( l \\), I'll divide both sides by 2:\n\\[\nl^3 = 900\n\\]\nTaking the cube root of both sides gives:\n\\[\nl = \\sqrt[3]{900} = 10\n\\]\nSo, the length of the prism is 10 units.\n</think>\n\nTo find the length of the rectangular prism, let's follow these steps:\n\n1. **Define the Variables:**\n \n - Let \\( l \\) be the length of the prism.\n - Since the length and width are equal, the width is also \\( l \\).\n - The height is twice the length, so the height is \\( 2l \\).\n\n2. **Write the Volume Formula:**\n \n The volume \\( V \\) of a rectangular prism is given by:\n \\[\n V = \\text{length} \\times \\text{width} \\times \\text{height}\n \\]\n Substituting the given values:\n \\[\n 1800 = l \\times l \\times 2l\n \\]\n Simplifying:\n \\[\n 1800 = 2l^3\n \\]\n\n3. **Solve for \\( l \\):**\n \\[\n 2l^3 = 1800\n \\]\n \\[\n l^3 = \\frac{1800}{2} = 900\n \\]\n \\[\n l = \\sqrt[3]{900}\n \\]\n \\[\n l = 10\n \\]\n\n4. **Final Answer:**\n \n The length of the prism is:\n \\[\n \\boxed{10}\n \\]"
}
] |
The probability it will rain on Saturday is $60\%$, and the probability it will rain on Sunday is $25\%$. If the probability of rain on a given day is independent of the weather on any other day, what is the probability it will rain on both days, expressed as a percent? | nvidia_math | [
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] | 0.926535 | 0.865388 | 0.061148 | <think>
Alright, so I have this probability problem here. It says that the probability it will rain on Saturday is 60%, and the probability it will rain on Sunday is 25%. They also mention that the probability of rain on a given day is independent of the weather on any other day. The question is asking for the probability it will rain on both days, expressed as a percent.
Hmm, okay. Let me try to figure this out step by step. I remember that when two events are independent, the probability of both happening is the product of their individual probabilities. Is that right? So, if Event A is one thing and Event B is another, and they don't affect each other, then P(A and B) = P(A) * P(B).
So, in this case, raining on Saturday is one event, and raining on Sunday is another event. Since they are independent, I can just multiply the two probabilities together to get the probability it rains both days.
Let me write that down to make sure I get it straight. The probability of rain on Saturday is 60%, which is 0.6 in decimal form. The probability of rain on Sunday is 25%, which is 0.25. So, the combined probability is 0.6 multiplied by 0.25.
Calculating that: 0.6 * 0.25. Hmm, 0.6 times 0.25, what is that? Well, 0.25 is a quarter, and a quarter of 0.6 is 0.15. So, that would be 0.15.
But wait, just to verify, let me multiply it out normally. 0.6 is like 6/10, and 0.25 is 25/100 or 1/4. So, multiplying 6/10 by 1/4:
6/10 * 1/4 = (6*1)/(10*4) = 6/40, which simplifies to 3/20.
3/20. Hmm, 3 divided by 20. Let me calculate that: 20 goes into 3 zero, point, 20 into 30 once (which is 0.1), 20 into 30 minus 20 is 10, bring down another 0: 100. 20 into 100 is 5 (so that's 0.15). So, 3/20 is 0.15. Yep, same as before.
So, 0.15 is 15%, right? 0.1 is 10%, 0.05 is 5%, so together it's 15%. So, a 15% chance it will rain on both Saturday and Sunday.
Let me think if there's any chance I can be mistaken here. The key here is that the events are independent, so one doesn't affect the other. Sometimes in probability problems, events can be dependent or independent, but this case specifically says it's independent, so we don't have to worry about the probability changing based on the other day.
So, just to recap: Probability it rains on Saturday is 60%, rain on Sunday 25%, independent, so we multiply the two probabilities to get the chance it rains both days, which is 0.6 * 0.25 = 0.15, so 15%.
Is there another way to represent this? Maybe using fractions converted from the percents. So, 60% is 3/5, 25% is 1/4. 3/5 * 1/4 is 3/20, which is indeed 15%, so that checks out the same result as before. So, 3/20 or 15%.
I think that's solid. I don't see another angle or consideration here. Maybe just reaffirming that independence is crucial here—if these weren't independent events, we'd have to consider conditional probabilities, but since they are independent, it's straightforward multiplication.
So, yes, I'm confident that 15% is the correct probability it will rain on both days.
**Final Answer**
The probability it will rain on both days is \boxed{15\%}.
</think>
The probability it will rain on Saturday is 60%, which is 0.6 in decimal form. The probability it will rain on Sunday is 25%, which is 0.25. Since the events are independent, the probability of both events occurring is the product of their individual probabilities.
To find the probability it will rain on both days, we multiply the two probabilities together:
\[
0.6 \times 0.25 = 0.15
\]
Converting this back to a percentage, we get 15%.
Thus, the probability it will rain on both days is \boxed{15\%}. | 12 | [
{
"from": "human",
"value": "The probability it will rain on Saturday is $60\\%$, and the probability it will rain on Sunday is $25\\%$. If the probability of rain on a given day is independent of the weather on any other day, what is the probability it will rain on both days, expressed as a percent?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I have this probability problem here. It says that the probability it will rain on Saturday is 60%, and the probability it will rain on Sunday is 25%. They also mention that the probability of rain on a given day is independent of the weather on any other day. The question is asking for the probability it will rain on both days, expressed as a percent.\n\nHmm, okay. Let me try to figure this out step by step. I remember that when two events are independent, the probability of both happening is the product of their individual probabilities. Is that right? So, if Event A is one thing and Event B is another, and they don't affect each other, then P(A and B) = P(A) * P(B).\n\nSo, in this case, raining on Saturday is one event, and raining on Sunday is another event. Since they are independent, I can just multiply the two probabilities together to get the probability it rains both days.\n\nLet me write that down to make sure I get it straight. The probability of rain on Saturday is 60%, which is 0.6 in decimal form. The probability of rain on Sunday is 25%, which is 0.25. So, the combined probability is 0.6 multiplied by 0.25.\n\nCalculating that: 0.6 * 0.25. Hmm, 0.6 times 0.25, what is that? Well, 0.25 is a quarter, and a quarter of 0.6 is 0.15. So, that would be 0.15.\n\nBut wait, just to verify, let me multiply it out normally. 0.6 is like 6/10, and 0.25 is 25/100 or 1/4. So, multiplying 6/10 by 1/4:\n\n6/10 * 1/4 = (6*1)/(10*4) = 6/40, which simplifies to 3/20.\n\n3/20. Hmm, 3 divided by 20. Let me calculate that: 20 goes into 3 zero, point, 20 into 30 once (which is 0.1), 20 into 30 minus 20 is 10, bring down another 0: 100. 20 into 100 is 5 (so that's 0.15). So, 3/20 is 0.15. Yep, same as before.\n\nSo, 0.15 is 15%, right? 0.1 is 10%, 0.05 is 5%, so together it's 15%. So, a 15% chance it will rain on both Saturday and Sunday.\n\nLet me think if there's any chance I can be mistaken here. The key here is that the events are independent, so one doesn't affect the other. Sometimes in probability problems, events can be dependent or independent, but this case specifically says it's independent, so we don't have to worry about the probability changing based on the other day.\n\nSo, just to recap: Probability it rains on Saturday is 60%, rain on Sunday 25%, independent, so we multiply the two probabilities to get the chance it rains both days, which is 0.6 * 0.25 = 0.15, so 15%.\n\nIs there another way to represent this? Maybe using fractions converted from the percents. So, 60% is 3/5, 25% is 1/4. 3/5 * 1/4 is 3/20, which is indeed 15%, so that checks out the same result as before. So, 3/20 or 15%.\n\nI think that's solid. I don't see another angle or consideration here. Maybe just reaffirming that independence is crucial here—if these weren't independent events, we'd have to consider conditional probabilities, but since they are independent, it's straightforward multiplication.\n\nSo, yes, I'm confident that 15% is the correct probability it will rain on both days.\n\n**Final Answer**\nThe probability it will rain on both days is \\boxed{15\\%}.\n</think>\n\nThe probability it will rain on Saturday is 60%, which is 0.6 in decimal form. The probability it will rain on Sunday is 25%, which is 0.25. Since the events are independent, the probability of both events occurring is the product of their individual probabilities.\n\nTo find the probability it will rain on both days, we multiply the two probabilities together:\n\n\\[\n0.6 \\times 0.25 = 0.15\n\\]\n\nConverting this back to a percentage, we get 15%.\n\nThus, the probability it will rain on both days is \\boxed{15\\%}."
}
] |
The side length of Square C is twice that of Square D. The area of Square D is $y$ square units. What is the area of Square C in terms of $y$? | tiger_math | [
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] | 0.921409 | 0.880603 | 0.040806 | <think>
To determine the area of Square C in terms of \( y \), I start by noting that Square C has a side length twice that of Square D. Since the area of Square D is \( y \), and area is calculated as the square of the side length, the side length of Square D is the square root of \( y \).
Next, the side length of Square C is twice that of Square D, which is \( 2 \times \sqrt{y} \). To find the area of Square C, I square this side length, resulting in \( (2\sqrt{y})^2 \). Calculating this gives \( 4y \), so the area of Square C is \( 4y \) square units.
</think>
To find the area of Square C in terms of \( y \), let's follow these steps:
1. **Understand the Relationship Between the Squares:**
- **Square C** has a side length that is **twice** the side length of **Square D**.
- Let the side length of Square D be \( s \). Therefore, the side length of Square C is \( 2s \).
2. **Express the Area of Square D:**
- The area of Square D is given by \( \text{Area}_D = s^2 \).
- According to the problem, \( \text{Area}_D = y \). Therefore:
\[
s^2 = y
\]
3. **Find the Area of Square C:**
- The side length of Square C is \( 2s \).
- The area of Square C is \( (2s)^2 \).
- Substitute \( s^2 \) from the previous step:
\[
(2s)^2 = 4s^2 = 4y
\]
4. **Final Answer:**
\[
\boxed{4y}
\] | 29 | [
{
"from": "human",
"value": "The side length of Square C is twice that of Square D. The area of Square D is $y$ square units. What is the area of Square C in terms of $y$?"
},
{
"from": "gpt",
"value": "<think>\nTo determine the area of Square C in terms of \\( y \\), I start by noting that Square C has a side length twice that of Square D. Since the area of Square D is \\( y \\), and area is calculated as the square of the side length, the side length of Square D is the square root of \\( y \\).\n\nNext, the side length of Square C is twice that of Square D, which is \\( 2 \\times \\sqrt{y} \\). To find the area of Square C, I square this side length, resulting in \\( (2\\sqrt{y})^2 \\). Calculating this gives \\( 4y \\), so the area of Square C is \\( 4y \\) square units.\n</think>\n\nTo find the area of Square C in terms of \\( y \\), let's follow these steps:\n\n1. **Understand the Relationship Between the Squares:**\n \n - **Square C** has a side length that is **twice** the side length of **Square D**.\n - Let the side length of Square D be \\( s \\). Therefore, the side length of Square C is \\( 2s \\).\n\n2. **Express the Area of Square D:**\n \n - The area of Square D is given by \\( \\text{Area}_D = s^2 \\).\n - According to the problem, \\( \\text{Area}_D = y \\). Therefore:\n \\[\n s^2 = y\n \\]\n\n3. **Find the Area of Square C:**\n \n - The side length of Square C is \\( 2s \\).\n - The area of Square C is \\( (2s)^2 \\).\n - Substitute \\( s^2 \\) from the previous step:\n \\[\n (2s)^2 = 4s^2 = 4y\n \\]\n\n4. **Final Answer:**\n\n \\[\n \\boxed{4y}\n \\]"
}
] |
"the length of a rectangle is reduced by 30 % . by what % would the width have to be increased to ma(...TRUNCATED) | tiger_math | [-0.004368847236037254,0.02906154841184616,-0.050959475338459015,-0.03280529007315636,0.044447451829(...TRUNCATED) | 0.931872 | 0.892841 | 0.039032 | "<think>\nTo solve this problem, I first acknowledge that the area of a rectangle is calculated by m(...TRUNCATED) | 28 | [{"from":"human","value":"the length of a rectangle is reduced by 30 % . by what % would the width h(...TRUNCATED) |
"Given two congruent cones with a radius of 12 cm and a height of 12 cm, enclosed within a cylinder,(...TRUNCATED) | tiger_math | [0.010961344465613365,0.04163513705134392,-0.059334851801395416,-0.03196265548467636,0.0225863382220(...TRUNCATED) | 0.912708 | 0.872674 | 0.040034 | "<think>\nAlright, let me try to figure this problem out step by step. So, we have two congruent con(...TRUNCATED) | 15 | [{"from":"human","value":"Given two congruent cones with a radius of 12 cm and a height of 12 cm, en(...TRUNCATED) |
"A bag contains 10 white balls and 5 black balls. A ball is drawn at random from the bag, and if it (...TRUNCATED) | nvidia_math | [0.006048833020031452,0.02329239621758461,-0.05593585595488548,-0.02552594058215618,0.03105423040688(...TRUNCATED) | 0.940686 | 0.884677 | 0.05601 | "<think>\nOkay, let me try to figure out this probability problem. So, there's a bag with 10 white b(...TRUNCATED) | 3 | [{"from":"human","value":"A bag contains 10 white balls and 5 black balls. A ball is drawn at random(...TRUNCATED) |
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