Question

13. The temperature on a heated metal plate positioned in the first quadrant of the zy-plane is given by C(x, y) = 25e -(0–1)2-(3-1) a Assume that temperature is measured in degrees Celsius and that x and y are y each measured in inches. a. Determine Czz(x, y) and Cyy(x, y). Do not do any additional work to algebraically simplify your results. b. Calculate Cz(1.1, 1.2). Suppose that an ant is walking past the point (1.1, 1.2) along the line y=1.2. Write a sentence to explain the meaning of the value of C. (1.1, 1.2), including units. C. Calculate Cyy(1.1, 1.2). Suppose instead that an ant is walking past the point (1.1, 1.2) along the line z=1.1. Write a sentence to explain the meaning of the value of Cyy(1.1, 1.2), including units. d. Determine Czy(x,y) and hence compute Czy(1.1, 1.2). What is the meaning of this value? Explain, in terms of an ant walking on the heated metal plate. =​

Answer 1

Answer: a)

C(x,y) = 25e^(-((x-1)^2 + (y-3)^2))

Czz(x,y) = ∂^2C/∂z^2 = 0 (since C does not depend on z)

Cyy(x,y) = ∂^2C/∂y^2 = 50e^(-((x-1)^2 + (y-3)^2))(2(y-3)^2 - 1)

b)

Cz(x,y) = ∂C/∂z = 0

Cz(1.1, 1.2) = 0

The value of Cz(1.1, 1.2) indicates the rate of change of temperature with respect to z at the point (1.1, 1.2), which is zero. In other words, the temperature does not change with respect to the height (z) at this point.

c)

Cyy(1.1, 1.2) = 50e^(-((1.1-1)^2 + (1.2-3)^2))(2(1.2-3)^2 - 1) ≈ -0.138

The value of Cyy(1.1, 1.2) indicates the rate of change of temperature with respect to y at the point (1.1, 1.2), which is approximately -0.138 °C per inch. In other words, as the ant moves along the line y=1.2, the temperature decreases by approximately 0.138 °C per inch.

d)

Czy(x,y) = ∂^2C/∂z∂y = 100e^(-((x-1)^2 + (y-3)^2))(y-3)

Czy(1.1, 1.2) = 100e^(-((1.1-1)^2 + (1.2-3)^2))(1.2-3) ≈ -43.44

The value of Czy(1.1, 1.2) indicates the rate of change of temperature with respect to both z and y at the point (1.1, 1.2), which is approximately -43.44 °C per inch per unit of height. In other words, as the ant moves along the line z=1.1 and simultaneously moves in the y-direction, the temperature decreases by approximately 43.44 °C per inch per unit of height.

Explanation: