Question

part b
if the initial height of the stone is decreased by
161.7 meters, how much less time in seconds will
it take the stone to reach the surface of the river?

1. the diagram below shows the dimensions of

adrians rectangular living room rug.
8 feet

he plans to buy a new rectangular rug that is
x feet longer and x feet wider.

part a
in terms of x, how many square feet of floor
space will adrians new rug cover?

part b
what value of x will result in a rug with an area
of 96 square feet?
a 4
b 8
c 12
d 16

Explanation:

Part A

Step1: Determine new length and width

The original length is 8 feet and width is 4 feet. The new length is \(8 + x\) feet and new width is \(4 + x\) feet.

Step2: Calculate area of new rug

The area \(A\) of a rectangle is length times width. So, \(A=(8 + x)(4 + x)\).
Expanding the product: \(A = 8\times4+8x + 4x+x^{2}=x^{2}+12x + 32\).

Step1: Set up the equation

From Part A, the area of the new rug is \(x^{2}+12x + 32\). We know the area is 96 square feet, so we set up the equation:
\(x^{2}+12x + 32 = 96\)

Step2: Simplify the equation

Subtract 96 from both sides: \(x^{2}+12x + 32-96 = 0\)
\(x^{2}+12x - 64 = 0\)

Step3: Solve the quadratic equation

We can factor the quadratic or use the quadratic formula. Let's try factoring. We need two numbers that multiply to -64 and add to 12. The numbers are 16 and -4.
So, \((x + 16)(x - 4)=0\)
Setting each factor equal to zero: \(x + 16 = 0\) or \(x - 4 = 0\)
\(x=-16\) (discarded since length can't be negative) or \(x = 4\)

Answer:

\(x^{2}+12x + 32\)

Part B