The head circumference C of a child is related to the height H of the child (both in inches) as: H(C) = 2.15C – 10.53 (a) Express the head circumference C as a function of height H. (b) Verify that C = C(H) is the inverse of H = H(C) by showing that H(C(H)) = H and C(H(C)) = C. (c) Predict the head circumference of a child who is 26 inches tall.  

Expert Answers

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The height H of a child as a function of the circumference of the head is given by H(C) = 2.15C – 10.53

Let y = H(C) = 2.15C – 10.53

=> 2.15C = y + 10.53

=> C = (y + 10.53)/2.15

From this we get C(H) = (H + 10.53)/2.15

 

H(C(H)) = H((H + 10.53)/2.15) = 2.15*((H + 10.53)/2.15) – 10.53

=> H + 10.53 - 10.53 = H

C(H(C)) = C(2.15C – 10.53) = (2.15C – 10.53 + 10.53)/2.15 = C

 

If the height of the child is 26 inches, the circumference of the head is C(26) = (26 + 10.53)/2.15 = 16.99 inches approximately.

 

The circumference of the head as a function of height is C(H) = (H + 10.53)/2.15 and the circumference for a height 26 inches is 16.99 inches approx.

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