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Chapter 3: Problem 33

The heat loss of a glass window varies jointly as the window's area and the difference between the outside and inside temperatures. A window 3 feet wide by 6 feet long loses 1200 Btu per hour when the temperature outside is \(20^{\circ}\) colder than the temperature inside. Find the heat loss through a glass window that is 6 feet wide by 9 feet long when the temperature outside is \(10^{\circ}\) colder than the temperature inside.

Short Answer

Expert verified
The heat loss through a glass window that is 6 feet wide by 9 feet long when the temperature outside is 10 degrees colder than the temperature inside is approximately 17982 Btu.

Step by step solution

01

Calculate Area and Compute Constant of Variation

Find the area of the first window by multiplying its width and length. Here, the width is 3 feet and length is 6 feet. So, the area, \(A_1 = 3 \times 6 = 18\) sq ft. Next, calculate the constant of variation, \(k\), using the formula \(k = y/(xz)\). In this case, \(y = 1200\) Btu (the heat loss), \(x = A_1 = 18\) sq ft (the area of the window) and \(z = 20^{\circ}\) (the temperature difference). So, we have \(k = 1200/(18 \times 20)\).
02

Compute the constant of variation

Compute the constant \(k\) using the mathematical operation determined in the first step. Therefore, \(k = 1200/(18 \times 20) = 3.33\) to two decimal places.
03

Calculate Area of Second Window

Find the area of the second window by multiplying its width and length. Here, the width is 6 feet and length is 9 feet. So, the area, \(A_2 = 6 \times 9 = 54\) sq ft.
04

Compute the Heat Loss of Second Window

Substitute the known values (\(k\), \(A_2\) and the new temperature difference) into the formula for joint variation to find the new heat loss. In this case, \(y = k \times x \times z = 3.33 \times 54 \times 10\).
05

Compute the Heat Loss

Perform the calculation outlined in step 4 to find the heat loss of the second window: \(y = 3.33 \times 54 \times 10 = 17982\) Btu to two decimal places.

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