Q. 90

Height and Head Circumference. The head circumference

C of a child is related to the height H of the child (both in inches) through the function $H (C) = 2.15 C - 10.53$

(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 $C (H (C)) = C$ .

Verified

Answer

Part (a). The head circumference is expressed in the terms of height as :

$C (H) = \frac{H + 10.53}{2.15}$ .

Part (b). It is verified that $C = C (H)$ is the inverse of $H = H (C)$ .

Part (c). The head circumference of a child who is $26$ inches tall is about $17$ inches.

1Part (a) Step 1. Express the function in terms of H.

The given function is $H (C) = 2.15 C - 10.53$

Adding $10.53$ to both sides, we get:

$H + 10.53 = 2.15 C - 10.53 + 10.53 H + 10.53 = 2.15 C$

Dividing both sides by $2.15$ , we get:

$\frac{H + 10.53}{2.15} = \frac{2.15 C}{2.15} C = \frac{H + 10.53}{2.15}$

2Part (b) Step 1. Verify the inverse

Substituting $C (H)$ in the function $H (C (H)) = H$ , we get:

$H (C (H)) = H H (\frac{H + 10.53}{2.15}) = H 2.15 (\frac{H + 10.53}{2.15}) - 10.53 = H H + 10.53 - 10.53 = H H = H$

This is true.

Substituting $H (C)$ in the function $C (H (C) = C$ , we get:

$C (H (C)) = C C (2.15 C - 10.53) = C \frac{2.15 C - 10.53 + 10.53}{2.15} = C \frac{2.15 C}{2.15} = C C = C$

This is true.

Thus, it is verified that $C = C (H)$ is the inverse of $H = H (C)$ .

3Part (c) Step 1. Find the circumference

Substituting $H = 26$ in the function $H (C) = 2.15 C - 10.53$ , we get:

$H = 2.15 C - 10.53 26 = 2.15 C - 10.53 26 +' 10.53 = 2.15 C 36.53 = 2.15 C \frac{2.15 C}{2.15} = \frac{36.53}{2.15} C = 16.99 C \approx 17$

Thus, the circumference is approximately $17 i n c h e s$