The Riverton High Rockets won their regional basketball tournament and were awarded this trophy.
A sphere of 9.5 inches in diameter is placed above the apex of the cone that has a height of 4.0 inches and it's diameter is 9.5 inches.
What is the approximate volume of the trophy?
A.
172.97
B.
206.74
C.
543.43
D.
732.45
B. 206.74
To find the volume of the trophy, we need to find the volumes of the sphere and the cone separately, and then add them together.
First, let's find the volume of the sphere using the formula:
Volume of sphere = (4/3)πr^3
Given that the diameter of the sphere is 9.5 inches, the radius can be calculated as half of the diameter:
Radius of sphere = (9.5 inches) / 2 = 4.75 inches
Substituting the value of the radius into the formula, we get:
Volume of sphere = (4/3) * π * (4.75 inches)^3
Calculating this, we get:
Volume of sphere ≈ 405.245 inches^3
Next, let's find the volume of the cone using the formula:
Volume of cone = (1/3)πr^2h
Given that the height of the cone is 4.0 inches and the diameter is 9.5 inches, we can first find the radius:
Radius of cone = (9.5 inches) / 2 = 4.75 inches
Substituting the values of the radius and height into the formula, we get:
Volume of cone = (1/3) * π * (4.75 inches)^2 * (4.0 inches)
Calculating this, we get:
Volume of cone ≈ 80.577 inches^3
Now, to find the approximate volume of the trophy, we add the volumes of the sphere and the cone:
Approximate volume of trophy ≈ Volume of sphere + Volume of cone
≈ 405.245 inches^3 + 80.577 inches^3
≈ 485.822 inches^3
Therefore, the approximate volume of the trophy is 485.822 inches^3.
The closest option to this answer is:
D. 732.45
To find the approximate volume of the trophy, we need to calculate the volume of the cone and the volume of the sphere, and then add them together.
First, let's find the volume of the cone using the formula for the volume of a cone: V = (1/3) * π * r^2 * h, where r is the radius and h is the height.
Given that the diameter of the cone is 9.5 inches, we can find the radius by dividing the diameter by 2: r = 9.5 / 2 = 4.75 inches.
Now we have all the values we need to find the volume of the cone: V_cone = (1/3) * π * (4.75)^2 * 4.0.
Using a calculator, we can calculate the volume of the cone as approximately 179.594 cubic inches.
Next, let's find the volume of the sphere using the formula for the volume of a sphere: V = (4/3) * π * r^3, where r is the radius.
Given that the diameter of the sphere is 9.5 inches, we can find the radius by dividing the diameter by 2: r = 9.5 / 2 = 4.75 inches.
Now we have all the values we need to find the volume of the sphere: V_sphere = (4/3) * π * (4.75)^3.
Using a calculator, we can calculate the volume of the sphere as approximately 356.519 cubic inches.
Finally, we can find the approximate volume of the trophy by adding the volumes of the cone and the sphere: V_trophy = V_cone + V_sphere.
V_trophy = 179.594 + 356.519.
Using a calculator, we can calculate the approximate volume of the trophy as approximately 536.113 cubic inches.
Comparing this value to the answer choices given:
A. 172.97
B. 206.74
C. 543.43
D. 732.45
The closest value is C. 543.43.
Therefore, the approximate volume of the trophy is approximately 543.43 cubic inches.