Question

a rancher has a roll of fencing to enclose a rectangular area. the table shows how the area that the rancher can enclose with the fencing depends on the width of the rectangle.

width w (ft)	area a (ft²)
20	1,600
30	2,100

which quadratic equation gives the area a of the rectangle in square feet given its width in w feet?
a(w)=w² + 90w
a(w)=-w² + 200w
a(w)=-w² + 100w
a(w)=w² + 40w

Explanation:

Step1: Recall area formula for rectangle

$A = lw$, and assume perimeter - related linear relationship for length $l$ in terms of width $w$. Let the general quadratic form of the area function be $A(w)=aw^{2}+bw + c$. Since when $w = 0$, $A=0$, then $c = 0$, so $A(w)=aw^{2}+bw$.

Step2: Substitute $w = 10$ and $A = 900$ into $A(w)=aw^{2}+bw$

$900=a\times10^{2}+b\times10=100a + 10b$. Divide by 10: $90 = 10a + b$.

Step3: Substitute $w = 20$ and $A = 1600$ into $A(w)=aw^{2}+bw$

$1600=a\times20^{2}+b\times20=400a+20b$. Divide by 20: $80 = 20a + b$.

Step4: Solve the system of equations

Subtract the equation from Step 2 from the equation from Step 3: $(20a + b)-(10a + b)=80 - 90$. This simplifies to $10a=-10$, so $a=-1$.

Step5: Find the value of $b$

Substitute $a=-1$ into $90 = 10a + b$. Then $90=10\times(-1)+b$, so $b = 100$.

Answer:

$A(w)=-w^{2}+100w$