2 part question
1) a cone shaped paper cup has a height of 7 inches and a diameter of 5 inches. What is the volume of the cone? Round answer to the nearest hundredth.
2) if we collect rainwater in the cup in question 1 until the waters depth is 6 inches, what percentage of the space inside the cup (volume) is still empty? Round answer to the nearest hundreth
1. V = pi*r^2*h/3.
V = 3.14(2.5)^2*7/3 = 45.81 In^3.
2. V = 3.14*(2.5)^2*1/3 = 6.54 In^3 empty.
%Empty = (6.54/45.81) + 100% = 14.28.
1) To find the volume of a cone, we can use the formula:
Volume = (1/3) × π × r² × h
where r is the radius of the base and h is the height of the cone.
Given the diameter of the cone is 5 inches, the radius can be calculated by dividing the diameter by 2:
Radius = 5 inches / 2 = 2.5 inches
Using the given height of 7 inches, we can plug these values into the formula to find the volume:
Volume = (1/3) × π × (2.5 inches)² × 7 inches
Calculating this gives us:
Volume = (1/3) × 3.14 × (2.5 inches)² × 7 inches
= (1/3) × 3.14 × 6.25 square inches × 7 inches
≈ 54.76 cubic inches
Rounding to the nearest hundredth, the volume of the cone is approximately 54.76 cubic inches.
2) If the water depth inside the cone is 6 inches, we need to calculate the volume of this water and compare it to the total volume of the cone.
The volume of water is given by the formula:
Volume of water = (1/3) × π × (2.5 inches)² × 6 inches
Calculating this gives us:
Volume of water = (1/3) × 3.14 × (2.5 inches)² × 6 inches
= (1/3) × 3.14 × 6.25 square inches × 6 inches
≈ 62.01 cubic inches
To find the percentage of space inside the cup that is still empty, we need to calculate the ratio of the remaining empty space to the total volume of the cone.
Empty space volume = Total volume of the cone - Volume of water
= 54.76 cubic inches - 62.01 cubic inches
≈ -7.25 cubic inches (Note: The result is negative because the volume of water exceeds the total volume of the cone.)
Since there is no empty space remaining, the percentage of space inside the cup that is still empty would be 0%.
To find the volume of a cone, you can use the formula:
Volume = (1/3) * π * r^2 * h
where r is the radius of the base and h is the height.
1) For the cone-shaped paper cup in question:
Given: Height (h) = 7 inches, Diameter = 5 inches
To find the volume, we need to determine the radius (r) of the base, which is half of the diameter. Thus, r = 2.5 inches.
Applying the formula:
Volume = (1/3) * π * (2.5^2) * 7
Using a calculator, the calculation becomes:
Volume ≈ 57.835 inches^3
Rounding to the nearest hundredth:
Volume ≈ 57.84 inches^3
Therefore, the volume of the cone-shaped paper cup is approximately 57.84 cubic inches.
2) If the depth of rainwater collected in the cup is 6 inches, we can calculate the percentage of space that is still empty by comparing the volume of the empty space with the total volume of the cup.
Empty space volume = Total volume of the cone - Volume of the water
Empty space volume = 57.84 inches^3 - Volume of the water
To find the volume of the water, we consider the smaller cone formed by the height of the water. The height in this case is 6 inches.
To find the radius of this smaller cone, we can use similar triangles:
(r / 7) = (2.5 / 6)
Solving for r, we get:
r = (2.5 * 7) / 6
r ≈ 2.92 inches
Now we can calculate the volume of the water using the formula:
Water volume = (1/3) * π * (2.92^2) * 6
Using a calculator, the calculation becomes:
Water volume ≈ 37.675 inches^3
Now we can calculate the percentage of empty space:
Percentage of empty space = (Empty space volume / Total volume) * 100
Plugging in the values:
Percentage of empty space = (57.84 inches^3 - 37.675 inches^3) / 57.84 inches^3 * 100
Using a calculator, the calculation becomes:
Percentage of empty space ≈ 34.82%
Therefore, approximately 34.82% of the space inside the cup is empty when the depth of the water is 6 inches.