Question

The diagram below shows a rectangular driveway ABCD and the road that the driveway connects to It ABH(5x-1) feet, ADe 9x feet and theperimeter of the rectangular driveway is (25x+1) feet.Da) The Aurora family plans to pick up a white plastic fenee around there three sides of their driveway. They are not putting a fence up wheretheir driveway meets the road. The plastic fence costs $4 per foot. How much money will the Aurora family spend putting up the fence?b) Next the Aurora family plans to repave the driveway Paving the drive costs $10 per square foot. How much money will the Aurora familyspend paving the driveway?

Answer 1

The rectangular driveway can be illustrated as shown below;

Note that for a rectangle, the opposite sides are equal in measure. Which means,

`\begin{gathered} AB=DC=(5x-1) \\ AD=BC=9x \end{gathered}`

Also, the perimeter of a rectangle is given by the formula;

`\text{Per}=2(l+w)`

Where;

`l=9x,w=(5x-1)`

Given the perimeter as (25x + 1), we would now have the following equation;

`\begin{gathered} \text{Per}=2(l+w) \\ 25x+1=2(9x+\lbrack5x-1\rbrack) \\ 25x+1=2(9x+5x-1) \\ 25x+1=2(14x-1) \\ 25x+1=28x-2 \\ We\text{ would now collect like terms and we'll have;} \\ 25x-28x=-2-1 \\ -3x=-3 \\ \text{Divide both sides by -3} \\ (-3x)/(-3)=(-3)/(-3) \\ x=1 \end{gathered}`

Next, the Aurora family plan to put up a plastic fence around their driveway. However the side that connects to the road would be excluded. This means side DC would be excluded when putting up the fence. Note that the sides AB and DC are equal in measure (opposite sides of a rectangle). Hence, when calculating the length of fence required, that would be the entire perimeter minus side DC.

That mean we would have the following;

`\begin{gathered} \text{Fence}=\text{Perimeter}-DC \\ \text{Fence}=(25x+1)-(5x-1) \\ \text{Note that ;} \\ x=1 \\ \text{Therefore;} \\ \text{Fence}=(\lbrack25*1\rbrack+1)-(\lbrack5*1\rbrack-1) \\ \text{Fence}=(25+1)-(5-1) \\ \text{Fence}=26-4 \\ \text{Fence}=22ft \end{gathered}`

We are also told that the fencing material costs $4 per foot. Now that the Aurora family would need 22 feet of plastic fence, the cost would be;

`\begin{gathered} \text{Cost}=\text{ \$4 per ft} \\ \text{Cost}=4(22) \\ \text{Cost}=88 \end{gathered}`

Therefore, the cost of fencing material would be $88.

The Aurora family next plans to pave the driveway. That entails covering the entire surface, in other words, the area of the rectangle.

The area of the rectangle is given by the formula;

`\text{Area}=l* w`

The length and width are as follows;

`\begin{gathered} l=9x \\ l=9*1 \\ l=9ft \\ \text{Also,} \\ w=(5x-1) \\ w=(\lbrack5*1\rbrack-1) \\ w=5-1 \\ w=4ft \end{gathered}`

The area therefore is;

`\begin{gathered} \text{Area}=l* w \\ \text{Area}=9*4 \\ \text{Area}=36ft^2 \end{gathered}`

Note that the cost of paving the driveway is given as $10 per square foot. The driveway however is 36 square feet. Hence the cost would be $10 times 36 square feet.

The cost therefore would be;

`\begin{gathered} \text{Cost}=\text{ \$10 per square foot} \\ \text{Cost}=10(36) \\ \text{Cost}=360 \end{gathered}`

Hence, the cost of paving the driveway is $360

ANSWER

x = 1

Part A: Cost = $88

Part B: Cost = $360