Question

These are the instructions. Write an equation to represent each situation. Use your equation to answer each question. Be sure to define your variables and identify which is independent and which is dependent. Show all work. Round any decimals to the nearest hundredth.

Answer 1

(a) Given x = 3, find the area of the deck.

Let:

Ad = Area of the deck

Ap = Area of the pool

At = Area total (deck + pool)

The area of the deck is:

Ad = At - Ap

So, let's find At and Ap.

Finding At:

At is the area of a rectangle whose measures are 10 + 2x and 14 + 2x.

Then: At = (10 + 2x)(14 + 2x)

Finding Ap:

Ap is the area of a rectangle whose measures are 10 and 14.

Then: Ap = 10*14 = 140

Then, Ad = At - Ap

Equation:

Ad = (10 + 2x)(14 + 2x) - 140

Depend variable = Ad

Independent variable = x

Finally, knowing that x = 3, let's find Ad.

Ad = (10 + 2x)(14 + 2x) - 140

Ad = (10 +2*3)(14 + 2*3) - 140

Ad = (10 + 6)(14 + 6) - 140

Ad = 16*20 - 140

Ad = 320 - 140

Ad = 180 square feet.

Answer: The area of the deck is 180 square feet.

(b) Given the area of the deck = 112 square feet, find x.

Let's use the equation from part A:

Ad = (10 + 2x)(14 + 2x) - 140

Now, Ad is known.

112 = (10 + 2x)(14 + 2x) - 140

Let's subtract 112 from both sides and multiply the values inside the parentheses:

`\begin{gathered} 112-112=(10+2x)(14+2x)-140-112 \\ 0=10*14+10*2x+2x*14+2x*2x-252 \\ 0=140+20x+28x+4x^2-252 \\ 0=4x^2+48x-112 \\ 4x^2+48x-112=0 \end{gathered}`

Now, let's solve the quadratic equation using the quadratic formula. For an equation ax²+ bx + c =0, the quadratic formula is:

`x=(-b\pm√(b^2-4ac))/(2a)`

In this question:

a = 4

b = 48

c = -112

Then:

Since x must be positive, x = 2 feet.

Answer: x = 2 feet.

In summary:

Equation:

Ad = (10 + 2x)(14 + 2x) - 140

Depend variable = Ad

Independent variable = x

(a) Ad = 180 square feet.

(b) x = 2 feet.