(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:
[tex]\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}[/tex]Now, let's solve the quadratic equation using the quadratic formula. For an equation ax²+ bx + c =0, the quadratic formula is:
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]In this question:
a = 4
b = 48
c = -112
Then:
[tex]\begin{gathered} x=\frac{-48\pm\sqrt{48^2-4*4*(-112)}}{2*4} \\ x=\frac{-48\pm\sqrt{2304+1792}}{2*4} \\ x=\frac{-48\operatorname{\pm}\sqrt{4096}}{8} \\ x=\frac{-48\pm64}{8} \\ x_1=\frac{-48-64}{8}=-14 \\ x2=\frac{-48+64}{8}=2 \end{gathered}[/tex]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.
