Page 32 - How To Solve Word Problems In Calculus

EXAMPLE 13
A river is 100 feet wide. The local telephone company wants
to run a cable from point A on one side of the river to a point
B on the other side, 500 feet downstream. It costs 3 dollars per
foot to run the cable under water while only 2 dollars per foot
to run the cable on land. Determine a function representing
the total cost to lay the cable.

Solution
Step1
Let x represent the number of feet from C, directly
opposite A, where the cable will emerge from the water, and
let y represent the number of feet of cable to be laid under
water.

500
C x 500 − x B

100 y

A

Step2
The total cost is the sum of the costs to run the cable on
land and under water.

C = C land + C water
= 2(500 − x) + 3y

= 1000 − 2x + 3y

Step3
√
2
2
By the Pythagorean theorem, y = x + 100 , so the total
cost function is

2
C(x) = 1000 − 2x + 3 x + 10,000 0 ≤ x ≤ 500
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