A poster has a perimeter of $16$ feet and an area of $15$ square feet. What are the dimensions of the poster? $\newline$ $\_\_\_\_$ feet by $\_\_\_\_$ feet

View step-by-step help

Home

Math Problems

Algebra 1

Area and perimeter: word problems

Full solution

Q. A poster has a perimeter of

16

feet and an area of

15

square feet. What are the dimensions of the poster?

\newline

\_\_\_\_

feet by

\_\_\_\_

feet

Define Variables: Let $l$ be the length and $w$ be the width of the poster. $\newline$ The perimeter of a rectangle is given by the formula $P = 2l + 2w$ .
Perimeter Equation: Given the perimeter of the poster is $16$ feet, we can write the equation $16 = 2l + 2w$ .
Simplify Perimeter: Simplify the equation by dividing all terms by $2$ to get $8 = l + w$ .
Area Equation: The area of a rectangle is given by the formula $A = lw$ . $\newline$ Given the area of the poster is $15$ square feet, we can write the equation $15 = lw$ .
Solve System of Equations: We now have a system of two equations with two variables: $\newline$ $1$ . $8 = l + w$ $\newline$ $2$ . $15 = lw$ $\newline$ We can solve this system of equations to find the values of $l$ and $w$ .
Express $w$ in terms of $l$ : From the first equation, we can express $w$ in terms of $l$ : $w = 8 - l$ .
Substitute into Area Equation: Substitute $w = 8 - l$ into the second equation: $15 = l(8 - l)$ .
Expand and Rearrange: Expand the equation: $15 = 8l - l^2$ .
Form Quadratic Equation: Rearrange the equation to form a quadratic equation: $l^2 - 8l + 15 = 0$ .
Factor Quadratic Equation: Factor the quadratic equation: $(l - 5)(l - 3) = 0$ .
Solve for l: Solve for l: $l = 5$ or $l = 3$ .
Find Dimensions: If $l = 5$ , then $w = 8 - l = 8 - 5 = 3$ . If $l = 3$ , then $w = 8 - l = 8 - 3 = 5$ .
Find Dimensions: If $l = 5$ , then $w = 8 - l = 8 - 5 = 3$ . If $l = 3$ , then $w = 8 - l = 8 - 3 = 5$ .We have two possible dimensions for the poster: $5$ feet by $3$ feet or $3$ feet by $5$ feet. Both sets of dimensions satisfy the given perimeter and area.