The area A of a shape is related to the length x of one side by the quadratic equation:?

Method 1 : By completing square

A = -7x² + 45x + 33
A = -7[x² - (45/7)x] + 33
A = -7[x² - 2*(45/14)x + (45/14)²] + 7*(45/14)² + 33
A = -7[x - (45/14)]² + (2949/28)

Since -7[x - (45/14)]² ≤ 0, then A = -7[x - (45/14)]² + (2949/28) ≤ 2949/28
Maximum area = 2949/28 = 110 sq. units (to 2 sig. fig.)


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Method 2 : By differentiation

A = -7x² + 45x + 33
dA/dx = -14x + 45
d²A/dx² = -14

When x = 45/14 :
dA/dx = 0 and d²A/dx² = -14 < 0
Hence, maximum A when x = 45/14

Maximum A = -7(45/14)² + 45(45/14) + 33 = 110 sq. units (to 2 sig. fig.)

A = -14x + 45
-14x + 45 = 0
x = 3.214
A_max = -7*(3.214)^2 + 45*3.214 + 33 = 105.32

A ` (x) = - 14x + 45 = 0 ____for max value of A
14 x = 45
x = 3 • 2
We agree !!!!!

Max occurs at -b/2a
-(45) / 2(-7) = 45/14 = 3 3/14 = 3.2142857142857...