Question

1. a magicians stage has a trapdoor.

a. the total area (in square feet) of the stage can be represented by $x^2 + 27x + 176$. write an expression for the width of the stage.
b. write an expression for the perimeter of the stage.
c. the area of the trapdoor is 10 square feet.
find the value of $x$.
d. the magician wishes to have the area of the stage be at least 20 times the area of the trapdoor.
does this stage satisfy his requirement? explain.
$2x$ ft
$(x + \frac{1}{2})$ ft
$(x + 16)$ ft

Explanation:

Step1: Factor area for width

We know area of a rectangle is $\text{length} \times \text{width}$. The length is given as $(x+16)$ ft, so we factor the area polynomial $x^2 + 27x + 176$:
$x^2 + 27x + 176 = (x+16)(x+11)$
Thus, width is $(x+11)$ ft.

Step2: Calculate perimeter

Perimeter of a rectangle is $2(\text{length} + \text{width})$. Substitute length $(x+16)$ and width $(x+11)$:
$2[(x+16)+(x+11)] = 2(2x+27) = 4x+54$

Step3: Solve for x using trapdoor area

Trapdoor area is given by $(x+\frac{1}{2})(2x) = 10$. First simplify the left side:
$2x(x+\frac{1}{2}) = 2x^2 + x$
Set equal to 10:
$2x^2 + x - 10 = 0$
Factor the quadratic:
$(2x+5)(x-2) = 0$
Since length cannot be negative, $x=2$.

Step4: Check stage area requirement

First calculate stage area when $x=2$:
$x^2 + 27x + 176 = 2^2 + 27(2) + 176 = 4 + 54 + 176 = 234$
20 times the trapdoor area is $20 \times 10 = 200$.
Since $234 > 200$, the stage meets the requirement.

Answer:

a. $\boldsymbol{x+11}$
b. $\boldsymbol{4x+54}$
c. $\boldsymbol{2}$
d. Yes, the stage satisfies the requirement. When $x=2$, the stage area is 234 square feet, which is greater than 20 times the trapdoor area (200 square feet).