Question

A rectangular room with a perimeter of 50 feet is to have an area of at least 120 square feet. Within what bounds must the length be?

Short answer

The length of the rectangle must be between 10 and 15 feet.

Step-by-step solution

Establish the relationship between length and width

The formula for the perimeter of a rectangle is $P=2l+2w$, with l representing the length and w the width of the rectangle. In this task, the perimeter is given as 50 feet. This can be solved to find the width in terms of length: $w=(P-2l)/2=25-l$.

Write the equation for the area

The area of a rectangle is given by $A=lw$. Substitute the expression for W in this formula to get the equation for the area as a function of L: $A(l)=l(25-l)$.

Determine the bounds for the length

The area must be at least 120 sq ft. Setting $A(l)\ge 120$ and solving the inequality for $l$, we find that $0<l<15$ and $10<l<25$. However, we need to find the intersection of these ranges to ensure both the perimeter and area conditions are met. The intersection is $10<l<15$.