The height of a certain hill (in feet) is given by where y is the distance (in miles) north, x...

Question

The height of a certain hill (in feet) is given by

where y is the distance (in miles) north, x the distance east of South Hadley.

(a) Where is the top of the hill located?

(b) How high is the hill?

(c) How steep is the slope (in feet per mile) at a point 1 mile north and one mile east of South Hadley? In what direction is the slope steepest, at that point?

Robert · Accepted Answer

(a) Say, hill top is located at (x∗, y∗). Then ∇h(x∗, y∗) = 0. This gives
2y∗ − 6x∗ − 18 = 0
2x∗ − 8y∗ + 28 = 0
}
x∗ = −2
y∗ = 3
The hilltop is at 3 miles north and 2 miles west of South Hadley.
(b) h(−2, 3) = 720ft .
(c) Putting in x = 1, y = 1, we get ∇h(1, 1) = 220(−x̂ + ŷ).

The height of a certain hill (in feet) is given by where y is the distance (in miles) north, x the distance east of South Hadley. (a) Where is the top of the hill located? (b) How high is the hill?...

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