ExpertHard4 marksStructured
AQA GCSE · Question 22 · Geometry and Measures
The metal used to make a sphere costs £4320. The metal costs £3.60 per gram. Each cubic centimetre of metal has a mass of 17.3 grams. Work out the radius, r, of the sphere. The volume of a sphere is given by V = (4/3)πr³.
The metal used to make a sphere costs £4320. The metal costs £3.60 per gram. Each cubic centimetre of metal has a mass of 17.3 grams. Work out the radius, r, of the sphere. The volume of a sphere is given by V = (4/3)πr³.
How to approach this question
1. This is a multi-step problem. Start by finding a value you can calculate from the given information. You have the total cost and the cost per gram. What can you find from this?
2. Once you have the total mass, use the density (mass per cm³) to find the total volume of the sphere.
3. Now you have the volume of the sphere. Use the given formula for the volume of a sphere to work backwards and find the radius, r.
4. Be careful when rearranging the formula to solve for r. You will need to multiply, divide, and finally take the cube root.
Full Answer
1. **Calculate the total mass of the metal.**
Mass = Total Cost / Cost per gram
Mass = £4320 / £3.60 per gram = 1200 grams.
2. **Calculate the volume of the metal (and the sphere).**
Volume = Total Mass / Mass per cm³
Volume = 1200 g / 17.3 g/cm³ = 69.36416... cm³.
3. **Use the volume formula to find the radius.**
Volume of a sphere, V = (4/3)πr³.
69.36416... = (4/3)πr³
Rearrange to solve for r³:
r³ = (69.36416... × 3) / (4π)
r³ = 208.0924... / (4π)
r³ = 16.561...
4. **Find r by taking the cube root.**
r = ³√16.561...
r = 2.549... cm.
Rounding to 3 significant figures, r = 2.55 cm.
This problem requires us to work backwards from the cost to find the radius.
Step 1: Find the total mass of the metal.
We know the total cost (£4320) and the cost per gram (£3.60).
Total Mass = Total Cost / Cost per gram
Total Mass = 4320 / 3.60 = 1200 g.
Step 2: Find the volume of the sphere.
We know the total mass (1200 g) and the density (17.3 g/cm³).
Volume = Total Mass / Density
Volume = 1200 / 17.3 = 69.36416... cm³.
It's important to keep this value unrounded in your calculator for the next step.
Step 3: Use the volume formula to find the radius.
The formula for the volume of a sphere is V = (4/3)πr³.
We substitute our calculated volume:
69.36416... = (4/3)πr³
Now, we rearrange to solve for r:
- Multiply by 3: 69.36416... × 3 = 4πr³ => 208.0924... = 4πr³
- Divide by 4π: r³ = 208.0924... / (4π) => r³ = 16.561...
Step 4: Find r.
To find r, we take the cube root of r³:
r = ³√(16.561...)
r = 2.549... cm
Rounding to 3 significant figures, the radius is 2.55 cm.
Common mistakes
✗ Mixing up the calculations, e.g., multiplying by density instead of dividing.
✗ Making an error when rearranging the volume formula, e.g., forgetting to multiply by 3 or divide by 4.
✗ Taking the square root instead of the cube root at the final step.
✗ Rounding prematurely, which can lead to an inaccurate final answer.
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