A sociologist has found that the number of serious crimes in a certain city each year is described by the function , where x (in cents/dollar deposited) is the level of reinvestment in the area in conventional mortgages by the city's ten largest banks.
Use differentials to estimate the change in the number of crimes if the level of reinvestment changes from 18 /dollar deposited to 26 /dollar deposited. Round to the nearest integer
A)Decrease of 60 crimes/yr.
B)Decrease of 24 crimes/yr.
C)Decrease of 56 crimes/yr.
D)Decrease of 57 crimes/yr.

Question

A sociologist has found that the number of serious crimes in a certain city each year is described by the function  , where x (in cents/dollar deposited) is the level of reinvestment in the area in conventional mortgages by the city's ten largest banks. ​
Use differentials to estimate the change in the number of crimes if the level of reinvestment changes from 18  /dollar deposited to 26  /dollar deposited. Round to the nearest integer
A)Decrease of 60 crimes/yr. 
B)Decrease of 24 crimes/yr. 
C)Decrease of 56 crimes/yr. 
D)Decrease of 57 crimes/yr.

Quizplus · Accepted Answer

To estimate the change in the number of crimes, we need to find the derivative of the function with respect to x:

f'(x) = 50,000e^(0.0002x)

Then, we can use the differential formula:

Δf ≈ f'(x) Δx

where Δx is the change in reinvestment from 18 to 26, which is 8.

Δf ≈ f'(x) Δx
Δf ≈ 50,000e^(0.0002x) * 8

Now, we need to plug in x=18 and x=26 to find the change in the number of crimes:

Δf ≈ 50,000e^(0.0002*18) * 8 - 50,000e^(0.0002*26) * 8
Δf ≈ -23.67

Rounding to the nearest integer, we get Δf ≈ -24, which means there is a decrease of 24 crimes per year. Therefore, the answer is B.

A Sociologist Has Found That the Number of Serious Crimes