Calculate the amount of material used in the construction of an igloo if its outer radius is $15$ feet and the thickness is $6$ inches.

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Q. Calculate the amount of material used in the construction of an igloo if its outer radius is

15

feet and the thickness is

6

inches.

Calculate Outer Volume: First, we need to determine the volume of the igloo with the outer radius before subtracting the volume of the igloo with the inner radius (which is the outer radius minus the thickness of the walls). The formula for the volume of a sphere (or in this case, a hemisphere since an igloo is half a sphere) is $V = \frac{4}{3}\pi r^3$ . However, since we only need the volume of a hemisphere, we will use $V = \frac{2}{3}\pi r^3$ .
Calculate Inner Volume: Calculate the volume of the igloo with the outer radius $15$ feet. Convert the radius to inches to match the thickness unit $6$ inches. There are $12$ inches in a foot, so the outer radius in inches is $15$ feet $* 12$ inches/foot $= 180$ inches. Now, calculate the volume using the formula $V = \frac{2}{3}\pi r^3$ . $\newline$ $V_{\text{outer}} = \frac{2}{3}\pi(180$ inches $)^3$
Subtract Volumes: Perform the calculation for the outer volume: $\newline$ $V_{\text{outer}} = \frac{2}{3}\pi(180^3)$ inches $^3$ $\newline$ $V_{\text{outer}} = \frac{2}{3}\pi(5,832,000)$ inches $^3$ $\newline$ $V_{\text{outer}} = \frac{2}{3}\pi(5,832,000)$ inches $^3$ $\newline$ $V_{\text{outer}} \approx \frac{2}{3} \times 3.14159 \times 5,832,000$ inches $^3$ $\newline$ $V_{\text{outer}} \approx 12,207,168\pi$ inches $^3$
Convert to Cubic Feet: Next, calculate the volume of the igloo with the inner radius. The thickness of the walls is $6$ inches, so the inner radius is $180$ inches $-$ $6$ inches $= 174$ inches. Now, calculate the volume using the formula $V = \frac{2}{3}\pi r^3$ . $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(174 \text{ inches})^3$
Convert to Cubic Feet: Next, calculate the volume of the igloo with the inner radius. The thickness of the walls is $6$ inches, so the inner radius is $180$ inches - $6$ inches = $174$ inches. Now, calculate the volume using the formula $V = \frac{2}{3}\pi r^3$ . $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(174 \text{ inches})^3$ Perform the calculation for the inner volume: $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(174^3) \text{ inches}^3$ $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(5,278,584) \text{ inches}^3$ $\newline$ $V_{\text{inner}} \approx \frac{2}{3} \times 3.14159 \times 5,278,584 \text{ inches}^3$ $\newline$ $V_{\text{inner}} \approx 11,048,314.32\pi \text{ inches}^3$
Convert to Cubic Feet: Next, calculate the volume of the igloo with the inner radius. The thickness of the walls is $6$ inches, so the inner radius is $180$ inches $- 6$ inches $= 174$ inches. Now, calculate the volume using the formula $V = \frac{2}{3}\pi r^3$ .
$V_{\text{inner}} = \frac{2}{3}\pi(174 \text{ inches})^3$ Perform the calculation for the inner volume:
$V_{\text{inner}} = \frac{2}{3}\pi(174^3) \text{ inches}^3$
$V_{\text{inner}} = \frac{2}{3}\pi(5,278,584) \text{ inches}^3$
$V_{\text{inner}} \approx \frac{2}{3} \times 3.14159 \times 5,278,584 \text{ inches}^3$
$V_{\text{inner}} \approx 11,048,314.32\pi \text{ inches}^3$ Now, subtract the inner volume from the outer volume to find the volume of the material used in the construction of the igloo.
$180$ $0$
$180$ $1$
Convert to Cubic Feet: Next, calculate the volume of the igloo with the inner radius. The thickness of the walls is $6$ inches, so the inner radius is $180$ inches $- 6$ inches $= 174$ inches. Now, calculate the volume using the formula $V = \frac{2}{3}\pi r^3$ . $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(174 \text{ inches})^3$ Perform the calculation for the inner volume: $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(174^3) \text{ inches}^3$ $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(5,278,584) \text{ inches}^3$ $\newline$ $V_{\text{inner}} \approx \frac{2}{3} \times 3.14159 \times 5,278,584 \text{ inches}^3$ $\newline$ $V_{\text{inner}} \approx 11,048,314.32\pi \text{ inches}^3$ Now, subtract the inner volume from the outer volume to find the volume of the material used in the construction of the igloo. $\newline$ $180$ $0$ $\newline$ $180$ $1$ Perform the subtraction to find the material volume: $\newline$ $180$ $1$ $\newline$ $180$ $3$
Convert to Cubic Feet: Next, calculate the volume of the igloo with the inner radius. The thickness of the walls is $6$ inches, so the inner radius is $180$ inches $- 6$ inches $= 174$ inches. Now, calculate the volume using the formula $V = \frac{2}{3}\pi r^3$ . $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(174 \text{ inches})^3$ Perform the calculation for the inner volume: $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(174^3) \text{ inches}^3$ $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(5,278,584) \text{ inches}^3$ $\newline$ $V_{\text{inner}} \approx \frac{2}{3} \times 3.14159 \times 5,278,584 \text{ inches}^3$ $\newline$ $V_{\text{inner}} \approx 11,048,314.32\pi \text{ inches}^3$ Now, subtract the inner volume from the outer volume to find the volume of the material used in the construction of the igloo. $\newline$ $180$ $0$ $\newline$ $180$ $1$ Perform the subtraction to find the material volume: $\newline$ $180$ $1$ $\newline$ $180$ $3$ To make the final answer more understandable, convert the material volume from cubic inches to cubic feet. There are $180$ $4$ inches in a foot, so there are $180$ $5$ cubic inches in a cubic foot. $\newline$ $180$ $6$ $\newline$ $180$ $7$
Convert to Cubic Feet: Next, calculate the volume of the igloo with the inner radius. The thickness of the walls is $6$ inches, so the inner radius is $180$ inches $- 6$ inches $= 174$ inches. Now, calculate the volume using the formula $V = \frac{2}{3}\pi r^3$ . $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(174 \text{ inches})^3$ Perform the calculation for the inner volume: $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(174^3) \text{ inches}^3$ $\newline$ $V_{\text{inner}} = \frac{2}{3}\pi(5,278,584) \text{ inches}^3$ $\newline$ $V_{\text{inner}} \approx \frac{2}{3} \times 3.14159 \times 5,278,584 \text{ inches}^3$ $\newline$ $V_{\text{inner}} \approx 11,048,314.32\pi \text{ inches}^3$ Now, subtract the inner volume from the outer volume to find the volume of the material used in the construction of the igloo. $\newline$ Material volume $180$ $0$ $\newline$ Material volume $180$ $1$ Perform the subtraction to find the material volume: $\newline$ Material volume $180$ $1$ $\newline$ Material volume $180$ $3$ To make the final answer more understandable, convert the material volume from cubic inches to cubic feet. There are $180$ $4$ inches in a foot, so there are $180$ $5$ cubic inches in a cubic foot. $\newline$ Material volume in cubic feet $180$ $6$ $\newline$ Material volume in cubic feet $180$ $7$ Perform the final calculation to convert the material volume to cubic feet: $\newline$ Material volume in cubic feet $180$ $7$ $\newline$ Material volume in cubic feet $180$ $9$ $\newline$ Material volume in cubic feet $- 6$ $0$ $\newline$ Material volume in cubic feet $- 6$ $1$