The surface of an air hockey table has an area of $40$ square feet and a perimeter of $28$ feet. What are the dimensions of the air hockey table? $\newline$ $\_\_\_$ feet by $\_\_\_$ feet

Full solution

Q. The surface of an air hockey table has an area of

40

square feet and a perimeter of

28

feet. What are the dimensions of the air hockey table?

\newline

\_\_\_

feet by

\_\_\_

feet

Define Area Formula: Let $l$ and $w$ be the length and width of the air hockey table, respectively. The area of a rectangle is given by $A = l \times w$ .
Substitute Area Value: Given area $A = 40$ square feet. Substitute into the area formula: $40 = l \times w$ .
Define Perimeter Formula: The perimeter of a rectangle is given by $P = 2l + 2w$ . Given perimeter $P = 28$ feet. Substitute into the perimeter formula: $28 = 2l + 2w$ .
Substitute Perimeter Value: Simplify the perimeter equation: $14 = l + w$ .
Simplify Perimeter Equation: We now have two equations: $40 = l \times w$ and $14 = l + w$ . Solve these equations simultaneously.
Solve Equations Simultaneously: Substitute $w = 14 - l$ into the area equation: $40 = l \times (14 - l)$ .
Substitute Width into Area Equation: Expand and rearrange the equation: $40 = 14l - l^2$ . Rearrange to form a quadratic equation: $l^2 - 14l + 40 = 0$ .
Expand and Rearrange Equation: Solve the quadratic equation using the quadratic formula, $l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Here, $a = 1$ , $b = -14$ , and $c = 40$ .
Solve Quadratic Equation: Calculate the discriminant: $(-14)^2 - 4 \cdot 1 \cdot 40 = 196 - 160 = 36$ .
Calculate Discriminant: Calculate the roots: $l = \frac{14 \pm \sqrt{36}}{2}$ . So, $l = \frac{14 \pm 6}{2}$ .
Calculate Roots: This gives $l = 10$ or $l = 4$ . If $l = 10$ , then $w = 14 - 10 = 4$ . If $l = 4$ , then $w = 14 - 4 = 10$ .