A prop for the theater club’s play is constructed as a cone topped with a half- sphere. What is the volume of the prop? Round yo
ur answer to the nearest tenth of a cubic inch. Use 3.14 to approximate pi, and make certain to show your work. Hint: you may need to find the volume of the component shapes
1 answer:
The volume of the prop is calculated to be 1,875.6 cubic cm.
Step-by-step explanation:
Step 1; The prop consists of a cone and a half-sphere on top. We will have to calculate the volumes of the cone and the half-sphere separately and then add them to obtain the total volume.
Step 2; The volume of a cone is determined by multiplying with π, the square of the radius (r²) and height (h). Here we substitute π as 3.14. The radius is 8 cm and the height is 12 cm.
The volume of the cone = × 3.14 × 8 × 8 × 12 = 803.84 cubic cm.
Step 3; The area of a half-sphere is half of a full sphere. The volume of a sphere is given by multiplying with π and the cube of the radius (r³). Here the radius is 8cm. We take π as 3.14.
The volume of a full sphere = × 3.14 × r³ =
× π × 8³ = 2,143.5733 cubic cm.
The volume of a half-sphere = = 1,071.7866 cubic cm.
Step 4; The total volume = The volume of the cone + The volume of the half sphere,
The total volume = 803.84 cubic cm + 1,071.7866 cubic cm = 1,875.6266 cubic cm. By rounding this off to the nearest tenth we get 1,875.6 cubic cm.
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<h3>Answer: roughly 12.6274 meters</h3>
The more accurate value is 12.6274169979696, though it's not fully exact.
Round this however you need to.
The exact distance 32*sin(45) - 10 meters.
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Explanation:
Refer to the diagram below.
I'll use point A in place of point O since the letter 'oh' is very similar looking to the number zero.
Plot point A at the origin (0,0). While at point A, look directly north. Then turn 30 degrees eastward to look at the bearing 030°. Next, move 20 meters along that bearing direction to arrive at point R. Segment AR is 20 meters long.
In the diagram, note how angle RAB is 30 degrees. The side opposite this is BR = m.
We can use the sine ratio to say that
sin(angle) = opposite/hypotenuse
sin(A) = BR/AR
sin(30) = m/20
m = 20*sin(30)
m = 10
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While still at point A, look directly north and turn 135 degrees clockwise (ie toward the east) and move 32 meters along that bearing. You'll arrive at point S as the diagram shows.
Notice how
angle RAB + angle RAC + angle CAS = 30+60+45 = 135
The remaining angle DAS is 180-135 = 45 degrees.
When focusing on triangle DAS, we can say
sin(A) = DS/AS
sin(45) = n/32
n = 32*sin(45)
n = 22.6274169979696
This value is approximate.
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Subtract the values of m and n
n - m = 32*sin(45) - 10 = exact distance
n - m = 22.6274169979696 - 10
n - m = 12.6274169979696
n - m = 12.6274 = approximate distance
Round it however you need to. I'm choosing to round to four decimal places.
So we see that point S is roughly 12.6274 meters east of point R.
If your teacher wants the exact distance, then stick with 32*sin(45)-10.