A toy manufacturer produces spherical balls of surface area 64π sq. in. How close to the ideal radius should the radius of the ball be, if the allowed error tolerance in the surface area is ±2 sq. in?

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01-01-2024
Mathematics

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A toy manufacturer produces spherical balls of surface area 64π sq. in. How close to the ideal radius should the radius of the ball be, if the allowed error tolerance in the surface area is ±2 sq. in?

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bbr7v6kzz7 bbr7v6kzz7 · Answer 1 · 2024-01-01T17:41:47+01:00

The surface area of a sphere is given by the formula 4πr^2, where r is the radius of the sphere.

Given that the surface area is 64π sq. in., we can equate the formula to the given surface area:
4πr^2 = 64π

Solving for r:
r^2 = 64π / 4π
r^2 = 16
r = √16
r = 4 inches

The ideal radius of the sphere should be 4 inches.

Now, to find the error tolerance, we can calculate the change in radius to achieve an error of ±2 sq. in. in the surface area.

The new surface area would be 64π + 2 and 64π - 2.

denote the ideal radius as r and the change in radius as δr.

For a change of +2 sq. in:
4π(r + δr)^2 = 64π + 2

For a change of -2 sq. in:
4π(r - δr)^2 = 64π - 2

Solving for δr in both cases will give us the required change in radius to achieve the specified error tolerance.