Question

A spherical fish bowl with radius 8 , is filled with water to a depth of 10 inches. The area of the surface of the water needs to be at least 150 square inches for proper aeration. Will this water depth satisfy that requirement? Justify your answer.

Answer 1

The water depth of 10 inches in the fishbowl satisfies the requirement for proper aeration with a surface area of approximately 502.65 square inches.

Step-by-step explanation:

To determine if the water depth in the fishbowl satisfies the requirement of a surface area of at least 150 square inches, we need to calculate the surface area of the water.

The surface area of the water in the fishbowl is the surface area of a spherical cap, which can be found using the formula:

A = 2πrh

Where A is the surface area, π is a constant (approximately 3.14159), r is the radius of the fishbowl, and h is the depth of the water.

Given that the radius of the fishbowl is 8 inches and the depth of the water is 10 inches, we can plug these values into the formula:

A = 2π(8)(10) = 160π ≈ 502.65 square inches

The surface area of the water in the fishbowl is approximately 502.65 square inches, which is greater than the required 150 square inches. Therefore, the water depth of 10 inches satisfies the requirement for proper aeration.