Let
x-------> the length of the placemat
w-----> the width of the placemat
we know that
area of the placemat=x*w
A=126 in²
so
A=x*w-----> equation 1
x=4*(9-0.5w)-----> equation 2
substitute equation 2 in equation 1
A=4*(9-0.5w)*w------>A=36w-2w²
the equation that models the area of the placemat, in terms of the width of the placemat, w is
A=-2w²+36w
A=126 in²
126=-2w²+36w
2w²-36w+126=0
using a graph tool----> to resolve the second order equation
see the attached figure
the solutions are
w1=4.76 in
w2=13.24 in
for w1=4.76 in
x=126/w----> x=126/4.76-----> x=26.47 in
for w1=13.24 in
x=126/w----> x=126/13.24-----> x=9.52 in
the answer is
the equation that models the area of the placemat, in terms of the width of the placemat, w is
A=-2w²+36w
