Question

1. find the dimensions of the rectangle. given that the area is 234 ft².

Explanation:

Step1: Recall area formula for rectangle

The area formula of a rectangle is $A = l\times w$, where $A$ is the area, $l$ is the length and $w$ is the width. Here, $A = 234$ square - feet, $l=x + 8$ and $w=x + 3$. So, $(x + 3)(x + 8)=234$.

Step2: Expand the left - hand side

Using the FOIL method, $(x + 3)(x + 8)=x^{2}+8x+3x + 24=x^{2}+11x + 24$. So the equation becomes $x^{2}+11x + 24=234$.

Step3: Rearrange to form a quadratic equation

Subtract 234 from both sides: $x^{2}+11x+24 - 234=0$, which simplifies to $x^{2}+11x - 210=0$.

Step4: Factor the quadratic equation

We need to find two numbers that multiply to - 210 and add up to 11. The numbers are 21 and - 10. So, $x^{2}+11x - 210=(x + 21)(x - 10)=0$.

Step5: Solve for x

Setting each factor equal to zero gives $x+21 = 0$ or $x - 10=0$. So, $x=-21$ or $x = 10$. Since the dimension of a rectangle cannot be negative, we take $x = 10$.

Step6: Find the dimensions

The length $l=x + 8=10 + 8=18$ feet and the width $w=x + 3=10+3 = 13$ feet.

Answer:

The dimensions of the rectangle are 13 feet and 18 feet.