Question

a rectangular deck is 12 ft by 14 ft. when the length and width are increased by the same amount, the area becomes 288 sq. ft. by how much were the dimensions increased?
○ 2 ft
○ 4 ft
○ 8 ft
○ 10 ft

Explanation:

Step1: Define the variable

Let \( x \) be the amount (in feet) by which both the length and the width are increased.

Step2: Write the area equation

The original length is \( 14 \) ft and width is \( 12 \) ft. After increasing, the new length is \( 14 + x \) and new width is \( 12 + x \). The area of a rectangle is length times width, so we have the equation:

$$ (14 + x)(12 + x) = 288 $$

Step3: Expand the left - hand side

Using the FOIL method: \( 14\times12+14x + 12x+x^{2}=288\)

$$ 168+26x + x^{2}=288 $$

Step4: Rearrange into standard quadratic form

Subtract \( 288 \) from both sides:

$$ x^{2}+26x + 168 - 288=0 $$

$$ x^{2}+26x-120 = 0 $$

Step5: Solve the quadratic equation

We can factor the quadratic equation. We need two numbers that multiply to \( - 120 \) and add up to \( 26 \). The numbers are \( 30 \) and \( - 4 \).

$$ x^{2}+30x-4x - 120=0 $$

$$ x(x + 30)-4(x + 30)=0 $$

$$ (x + 30)(x - 4)=0 $$

Setting each factor equal to zero:
\(x+30 = 0\) gives \(x=-30\) (we discard this solution because a length cannot be increased by a negative amount)
\(x - 4=0\) gives \(x = 4\)

Answer:

4 ft (the option is "4 ft")