The rate of change in the number of miles of road cleared per hour by a snowplow with respect to the depth of the snow is inversely proportional to the depth of the snow. Given that 20 miles per hour are cleared when the depth of the snow is 2.5 inches and 11 miles per hour are cleared when the depth of the snow is 7 inches, then how many miles of road will be cleared each hour when the depth of the snow is 13 inches

5.6 miles

Step-by-step explanation:

From the question, we can infer that this is an inversely proportionality question.

Let us represent Miles per hour = N

Depth of snow = s

Where k is proportionality constant.

dN/ds = k/s

dN = ds .k/s

Integrating this

∫dN = ∫k/s ds

N = k In s + C

Step 1

We find k

From the question, we are told that:

Given that 20 miles per hour are cleared when the depth of the snow is 2.5 inches

11 miles per hour are cleared when the depth of the snow is 7 inches,

20 = k ln 2.5 + C Equation 1

11 = k ln 7 + C Equation 2

11 = k ln 7 + 20 - k ln 2.5

Collect like terms

k ln 2.5 - k ln 7 = 20 - 11

k( In 2.5 - In 7) = 9

k = 9/( In 2.5 - In 7)

k = 9/ -1.0296194172

k = -8.7410938932

Step 2

We find C

20 = k ln 2.5 + C Equation 1

C = 20 - k ln 2.5

C = 20 -(-8.7410938932 In 2.5)

C = 20 + 8.7410938932 In 2.5

C = 20 + 8.0093833208

C = 28.0093833208

Step 3

how many miles of road will be cleared each hour when the depth of the snow is 13 inches

N = k In s + C

= -8.7410938932 ln 13 + 28.0093833208

= -22.420463165 + 28.0093833208

= 5.5889201559

Approximately= 5.6 miles

Therefore, 5.6 miles of road will be cleared each hour when the depth of the snow is 13 inches.