Question

The price of a floor tile is $3 per square foot, and the cost of shipping is $20. You are adjusting the size of your bathroom according to the space availability. You would like to estimate the rate of change of cost for flooring that you expect when you change the size of the square bathroom of current length 10 feet. Assume xx to be the length of a square shaped bathroom. Include units in all parts. a. Calculate the average rate of change of cost when you change your bathroom from 10 feet to 12 feet long. b. Calculate the instantaneous rate of change of cost at length 10 feet. c. If the cost of shipping also depends on the length according to \$20(1 \sin{x})$20(1 sinx), what is the instantaneous change of cost at length 10 feet.

Answer 1

a

`A =  \$66 \foot`

b

tex]C(10)' = \$60 / feet[/tex]

c

`C(10)' = \$79.696 \ feet`

Explanation:

From the question we are told that

The price per square foot of a floor tile is
`p = \$ 3`

The cost of shipping is
`s =  \$ 20`

The current length of the square bathroom is x = 10 ft

The new length is
`x_ 1 =  12 \  ft`

Generally the equation representing the cost of the square tile is mathematically represented as

`C(x) =  3x^2 + 20`

Generally the average rate of change of cost is mathematically represented as

`A =  ( cost \ for\  a \  12 feet \  bathroom - cost \ for\  a \  10 feet \  bathroom  )/(12 feet  - 10 feet)`

Generally the cost for flooring a 12 feet square bathroom is mathematically represented as

`C(12) =  3(12)^2 + 20`

=>
`C(12) = \$ 452`

Generally the cost for flooring a 10 feet square bathroom is mathematically represented as

`C(10) =  3(10)^2 + 20`

=>
`C(10) =  \$320`

So

`A =  ( 452- 320 )/(12 feet  - 10 feet)`

=>
`A =  \$66 \foot`

Generally the instantaneous rate of change of cost is obtained by the differentiating the cost function as follows

`C(x) =  3x^2 + 20`

`C(x)' = (dC(x))/(dx)  =  6x`

So the instantaneous rate of change of cost at length 10 feet is mathematically represented as

`C(10)' = 6(10)`

=>
`C(10)' = \$60  /  feet`

Generally given that the new cost function is

`C(x) =  20(1 + sinx) + 3x^2`

Now the instantaneous rate of change of cost will now be

`C(x)' =  20cos(x) + 6x`

So the instantaneous rate of change of cost at length 10 feet is mathematically represented as

`C(10)' =  20cos(10) + 6* 10`

=>
`C(10)' = \$79.696 \ feet`