Question

The heights of two different projectiles after they are launched are modeled by f(x) and g(x). The function f(x) is defined as f(x) = -1672 +42x + 12. The table contains the values for the quadratic function g. x, G(x): 0,9 1,33 2,25 What is the approximate difference in the maximum heights achieved by the two projectiles?
5.6 feet 5.4 feet 3.0 feet
0.2 feet

Answer 1

The approximate difference in the maximum heights achieved by the two projectiles is 3.7 feet .

Given that :

f(x) = -16x^2 + 42x + 12,

f'(x) = -32x + 42 = 0 ,

so x = 42/32

= 21/16

= 1.3125

Then its max height is:

f(x) = f(21/16) = -16(21/16)2 + 42(21/16) + 12

f(x) = [-(21)2 + 42(21) + 12(16)]/16

= [-441 + 882 + 192]/16 = 633/16

= 39.563

= 39.6 rounded

g(0) =  -9 =  C

g(1) =  33 =  A + B + C = A + B -9,  so A = 33 + 9 -B = (42 - B)

g(2) =  25 = 4A + 2B - 9

25 = 4(42 - B) + 2B -9 = (168 - 9) -2B,

so  2B = -25 + 159 = 134  

B = 67

Then A = 42 - B

= 42 - 67

= - 25

Thus g(x) = -25x2 + 67x - 9,

and its derivative or slope is g'(x) = -50x + 67 = 0 for the peak height

x  = -67/-50 = 1.34,

g(x) = -25(1.34)2 + 67(1.34) - 9

= -44.89 + 89.78 - 9

= 35.89

= 35.9 rounded

Finally, the difference in peak or max height = 39.6 - 35.9

=  3.7 feet

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