Question

attempt 1: 10 attempts remaining. a carpenter is building a rectangular shed with a fixed perimeter of 42 feet. find the dimensions of the largest floor area of the shed that can be built and find the maximum floor area of the shed. side length:? maximum area:?

Explanation:

Step1: Recall rectangle perimeter and area formulas

For a rectangle with length \( l \) and width \( w \), the perimeter \( P = 2(l + w) \) and area \( A = lw \). Given \( P = 42 \) feet, we have \( 2(l + w)=42 \), which simplifies to \( l + w = 21 \), so \( w = 21 - l \).

Step2: Express area in terms of one variable

Substitute \( w = 21 - l \) into the area formula: \( A(l)=l(21 - l)=21l - l^{2} \). This is a quadratic function in the form \( A(l)=-l^{2}+21l \), where \( a=-1 \), \( b = 21 \), \( c = 0 \).

Step3: Find the vertex of the quadratic function

For a quadratic function \( f(x)=ax^{2}+bx + c \), the \( x \)-coordinate of the vertex (which gives the maximum for \( a<0 \)) is \( x=-\frac{b}{2a} \). Here, \( l=-\frac{21}{2(-1)}=\frac{21}{2}=10.5 \) feet. Then \( w = 21 - 10.5 = 10.5 \) feet (so it's a square, which makes sense for maximum area of a rectangle with fixed perimeter).

Step4: Calculate the maximum area

Substitute \( l = 10.5 \) and \( w = 10.5 \) into the area formula: \( A=(10.5)(10.5)=110.25 \) square feet.

Answer:

Side length (both length and width, since it's a square): \( 10.5 \) feet
Maximum area: \( 110.25 \) square feet