Question

The demand function for a product is given by p = 5000 ln(x + 10) where p is the price per unit in dollars when x units are demanded. (a) Find the rate of change of price with respect to the number of units sold when 40 units are sold. (Round your answer to the nearest cent.) $ (b) Find the rate of change of price with respect to the number of units sold when 90 units are sold. (Round your answer to the nearest cent.) $ (c) Find the second derivative. p''(x) = Is the rate at which the price is changing at 40 units increasing or decreasing? p'(x) is increasing at 40 units. p'(x) is decreasing at 40 units.

Answer 1

(a)The rate of change of price to the number of units sold when 40 units are sold is -$5.28.(decrease)

(b)The rate of change of price to the number of units sold when 40 units are sold is -$1.89.(decrease)

(c)The p' is decreasing at 40 units.

Explanation:

Given function is

`p=(5000)/(ln(x+10))`

where p is the price per unit dollar when x units are demanded.

`p=(5000)/(ln(x+10))`

Differentiating with respect to x

`p'=(d)/(dx)((5000)/(ln(x+10)))`

`\Rightarrow p'=((d)/(dx)(4000).(ln(x+10))-4000.(d)/(dx)(ln(x+10)))/((ln(x+10))^2)`

`\Rightarrow p'=- (4000)/((x+10)(ln(x+10))^2)`

Again differentiating with respect to x

`p''=-((d)/(dx) (4000).(x+1) (ln (x+10)) ^2- 4000.(d)/(dx) (x+1) (ln (x+10))^2)/((x+1)^2 (ln (x+10))^4)`

`\Rightarrow p''=(4000[(ln(x+10))^2.1+(x+10).(2(ln(x+10)))/((x+10))])/((x+10)^2(ln(x+10))^4)`

`\Rightarrow p''=\frac{4000[(ln(x+10))^2+{2(ln(x+10))}]}{(x+10)^2(ln(x+10))^4}`

`\Rightarrow p''=(4000[(ln(x+10))+2])/((x+10)^2(ln(x+10))^3)`

(a)

Now,

`\left p'\right|_(x=40)=- (4000)/((40+10)(ln(40+10))^2)`

=- $5.28

The rate of change of price to the number of units sold when 40 units are sold is - $5.28.

(b)

`\left p'\right|_(x=90)=- (4000)/((90+10)(ln(90+10))^2)`

= -$1.89

The rate of change of price to the number of units sold when 40 units are sold is -$1.89.

(c)

`p''|_(x=40)=(4000[(ln(40+10))+2])/((40+10)^2(ln(40+10))^3)>0`

Since p''>0.

The rate is decreasing at 40 units.