A magician's stage has a trap door.
a. The total area of the stage can be represented by x^2++27x+176. Write an expression for the width of the stage.
b. The area of the trap door is 12 suare feet. Find the value of x.
c. What fraction of the area of the stage is the area of the trap door?
Trap door: x ft by (x-1) ft
Stage: (x+16) ft
Nuts
a. To find the expression for the width of the stage, we need to factor the expression x^2 + 27x + 176 into the form (x + a)(x + b), where a and b are the dimensions of the stage.
The width of the stage represents one of the factors. In this case, we need to find the factors of 176 that add up to 27. After factoring, we have:
x^2 + 27x + 176 = (x + 16)(x + 11)
Thus, the width of the stage is (x + 16) ft.
b. The area of the trap door is given as 12 square feet, which means the product of its dimensions (x ft by (x-1) ft) is equal to 12.
(x)(x-1) = 12
Expanding this equation, we have:
x^2 - x - 12 = 0
We can solve this quadratic equation by factoring or using the quadratic formula. Factoring, we have:
(x - 4)(x + 3) = 0
This gives us two possible solutions: x = 4 or x = -3. Since the width of the stage can't be negative, we discard x = -3.
Therefore, the value of x is 4.
c. To find the fraction of the area of the stage that is occupied by the trap door, we need to compare the area of the trap door to the total area of the stage.
The area of the trap door is given as 12 square feet, which is the product of its dimensions (x ft by (x-1) ft).
The width of the stage, as we found in part a, is (x + 16) ft. The length of the stage is (x + 11) ft.
Thus, the area of the stage is given by:
Area of the stage = (x + 16)(x + 11)
To find the fraction, we divide the area of the trap door by the area of the stage:
Fraction = Area of the trap door / Area of the stage
Fraction = 12 / [(x + 16)(x + 11)]
I don't get the concept
x^2+27x+176 = (x+16)(x+11)
x(x-1)=12
x^2-x-12=0
(x-4)(x+3)=0
Assuming the same value of x, then
trap/stage = 12/(x^2+27x+176)