Given that the volume of a sphere is 562.5𝜋 cubic centimeters, find its surface area in terms of 𝜋.
So in this question, we’re told the volume of the sphere. The volume is the amount of space that the object takes up. We’re asked to find the surface area. That is the area of the flat surface around the outside of the sphere. To answer this question, we’re going to use two formulas, the volume and the surface area.
The volume of a sphere is equal to four-thirds times 𝜋 times 𝑟 to the third power. The surface area of a sphere is equal to four times 𝜋 times 𝑟 squared. In both of these, the 𝑟 refers to the radius of the sphere. Let’s draw the radius of our sphere on the diagram. We don’t know a numerical value for the radius, but we can call it the letter 𝑟 to represent it.
Now, we’re going to find the radius 𝑟 by using the calculation for the volume. We can start by writing out the formula for the volume of a sphere which is four-thirds times 𝜋 times 𝑟 to the third power. And we can plug in the value for the volume, 562.5𝜋, into this formula, which gives us 562.5𝜋 equals four-thirds 𝜋𝑟 to the third power. As we have 𝜋 on both sides of our equation, we can divide through by 𝜋. Which is 562.5 equals four-thirds 𝑟 to the third power.
In order to work towards getting 𝑟 by itself, we could see that on the right-hand side, we have four-thirds times 𝑟 to the third power. In order to apply the inverse operation, we would divide by four-thirds. And when we’re dividing by a fraction, we change the division to multiplication and flip the numerator and denominator. So, dividing by four over three is the same as multiplying by three-quarters. So, we have three-quarters times 562.5 equals 𝑟 to the third power. Evaluating this using a calculator will give us 421.875 equals 𝑟 to the third power.
Next, to find 𝑟 by itself, we do the inverse operation of finding the third power. And that’s to take the cube root of both sides. So, 𝑟 is equal to the cube root of 421.875, which we can evaluate as 7.5. This means that the radius of our sphere, 𝑟, is equal to 7.5 centimeters. And now, we can use the radius value to find the surface area of the sphere. And we start by writing the formula, the surface area of the sphere equals four times 𝜋 times 𝑟 squared. Substituting our value of 7.5 for 𝑟 will give us four times 𝜋 times 7.5 squared.
It’s important when we’re doing this calculation that we notice it’s only the 7.5 that is squared and not all the other values as well. So, evaluating 7.5 squared then, we get a right-hand side of four times 𝜋 times 56.25. We can then multiply the four and the 56.25 to give us 225𝜋.
At this point, we might go ahead and use the 𝜋 button on our calculator or the value 3.14 and multiply it with 225 to get a value. However, the question has asked us to leave our answer in terms of 𝜋. So, the answer, 225𝜋, is all that’s required. And since this is an area calculation, the units here will be square centimeters. So, the final answer for the surface area of the sphere is 225𝜋 square centimeters.