Question

The spheres are similar. The surface area of the blue sphere is 4071.5 square inches. What is the surface area of the red sphere? Round your answer to the nearest tenth please help me this is my last test before spring break

Answer 1

Therefore, the surface area of the red sphere is approximately 373.7 square inches, rounded to the nearest tenth.

Explanation:

Since the spheres are similar, their surface areas are proportional to the square of their radii.

Let r be the radius of the red sphere, and let R be the radius of the blue sphere. Then:

S_red / S_blue = (r / R)²

We know that S_blue = 4071.5, and we know that the blue sphere has a radius of 24 inches. Therefore:

S_red / 4071.5 = (r / 24)²

To find S_red, we need to solve for r:

(r / 24)² = S_red / 4071.5

r / 24 = sqrt(S_red / 4071.5)

r = 24 * sqrt(S_red / 4071.5)

Now we can substitute this expression for r into the equation S_red / S_blue = (r / R)²:

S_red / 4071.5 = (24 * sqrt(S_red / 4071.5) / R)²

S_red / 4071.5 = (576 * S_red / (4071.5 * R²))

S_red = (576 / 4071.5) * S_red * R²

Dividing both sides by S_red and simplifying:

1 = 576 / 4071.5 * R²

R² = 576 / 4071.5

R = sqrt(576 / 4071.5) = 0.3051

So the radius of the blue sphere is approximately 0.3051 times the radius of the red sphere. Therefore:

S_red / S_blue = (R / r)² = (1 / 0.3051)² = 10.89

S_red = 4071.5 / 10.89 = 373.7

Therefore, the surface area of the red sphere is approximately 373.7 square inches, rounded to the nearest tenth.