The height of a hill (in feet) is given by where x is the distance (in miles) east and y is the distance (in miles) north of your cabin. If you are at a point on the hill 1 mile north and 1 mile east of your cabin, what is the rate of change of the height of the hill (a) in a northerly direction and (b) in an easterly direction?
A)(a) 570 ft/mi, (b) 690 ft/mi
B)(a) -570 ft/mi, (b) 690 ft/mi
C)(a) 690 ft/mi, (b) 570 ft/mi
D)(a) 690 ft/mi, (b) -570 ft/mi

Question

The height of a hill (in feet) is given by  where x is the distance (in miles) east and y is the distance (in miles) north of your cabin. If you are at a point on the hill 1 mile north and 1 mile east of your cabin, what is the rate of change of the height of the hill (a) in a northerly direction and (b) in an easterly direction?
A)(a) 570 ft/mi, (b) 690 ft/mi 
B)(a) -570 ft/mi, (b) 690 ft/mi 
C)(a) 690 ft/mi, (b) 570 ft/mi 
D)(a) 690 ft/mi, (b) -570 ft/mi

Quizplus · Accepted Answer

(a) To find the rate of change of the height of the hill in a northerly direction, we need to differentiate the height function with respect to y (since we are moving in the y direction).

11eaa3e5_54e9_d40b_9f8f_1dd443436cc1_TB5971_11 = 11(x^2 + y^2)^(3/2)

Differentiating with respect to y gives:

(d/dy) 11(x^2 + y^2)^(3/2) = 33y(x^2 + y^2)^(1/2)

Plugging in x = 1 and y = 1 (since we are 1 mile north and 1 mile east of our cabin) gives:

33(1)(2)^(1/2) = 66^(1/2) ≈ 8.12

Therefore, the rate of change of the height of the hill in a northerly direction is 8.12 ft/mile.

(b) To find the rate of change of the height of the hill in an easterly direction, we need to differentiate the height function with respect to x (since we are moving in the x direction).

Differentiating with respect to x gives:

(d/dx) 11(x^2 + y^2)^(3/2) = 33x(x^2 + y^2)^(1/2)

Plugging in x = 1 and y = 1 gives:

33(1)(2)^(1/2) = 66^(1/2) ≈ 8.12

Therefore, the rate of change of the height of the hill in an easterly direction is 8.12 ft/mile.

However, since we are moving in the positive x direction, the rate of change of the height of the hill should be positive as well. This means that the correct answer choice is B, which has a negative value for the rate of change of the height of the hill in the northerly direction.

The Height of a Hill (In Feet) Is Given by