Video: Forming Inverse Functions Involving Geometric Formulas

The surface area, 𝐴, of a sphere in terms of its radius, 𝑟, is given by 𝐴(𝑟) = 4𝜋𝑟². Express 𝑟 as a function of 𝐴 and find, to the nearest tenth of an inch, the radius of a sphere whose surface area is 1,000 square inches.

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Video Transcript

The surface area, 𝐴, of a sphere in terms of its radius, 𝑟, is given by 𝐴 of 𝑟 is equal to four 𝜋𝑟 squared. Express 𝑟 as a function of 𝐴 and find, to the nearest tenth of an inch, the radius of a sphere whose surface area is 1,000 square inches.

We are told in the question that the surface area of a sphere 𝐴 is equal to four 𝜋𝑟 squared. In order to express 𝑟 as a function of 𝐴, we need to rearrange the formula to make 𝑟 the subject. We can do this in two steps, firstly, by dividing both sides of the equation by four 𝜋. This gives us 𝐴 over four 𝜋 is equal to 𝑟 squared. As the inverse or opposite of squaring is square rooting, we need to square root both sides of this equation. The square root of 𝐴 over four 𝜋 is equal to 𝑟. As the radius 𝑟 is a length, this answer must be positive.

The second part of our question asks us to calculate the radius of a sphere whose area is 1,000 square inches. We can do this by substituting 𝐴 equals 1,000 into our new formula. This gives us 𝑟 is equal to the square root of 1,000 over or divided by four 𝜋. Typing this into the calculator gives us 𝑟 is equal to 8.9206 and so on. We need to round our answer to the nearest tenth of an inch. This is the same as rounding to one decimal place. As the two is less than five, we round down so that 𝑟 is equal to 8.9 inches. This is the radius of a sphere whose surface area is 1,000 square inches.

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