Question

The temperature in degrees Celsius on the surface of a metal plate is

T(x, y) = 40 − 6x2 − y2
where x and y are measured in centimeters. Estimate the average temperature when x varies between 0 and 3 centimeters and y varies between 0 and 4 centimeters.

Answer 1

The student's question concerns finding the average temperature over a range of x and y coordinates on a metal plate, described by T(x, y) = 40 − 6x^2 − y^2. The solution involves double integration over the rectangle bounded by x=0 to 3 cm and y=0 to 4 cm, followed by dividing the result by the area of the rectangle.

Step-by-step explanation:

The student is asking to estimate the average temperature on the surface of a metal plate, described by the temperature function T(x, y) = 40 − 6x2 − y2, within a specific range for x and y coordinates measured in centimeters. To find this average temperature, we can use double integration over the given range for x (0 to 3 cm) and y (0 to 4 cm).

Firstly, calculate the integral of the temperature function over the given range for x and y:

1. Integrate T(x, y) with respect to x from 0 to 3.
2. Integrate the result with respect to y from 0 to 4.

This will give us the total temperature over the area. Then, the average temperature is found by dividing this total by the area of the rectangle (3 cm × 4 cm).