Question

Assume the weight (in grams) of the human brain during the last trimester
of gestation and the first two years after birth can be approximated by the
function
𝑤 = 𝑐3
110 − 1500
𝑐 ,
where c is the circumference of the head in cm. Find the approximate
weight of brains with a circumference of 30 or 40 cm

Answer 1

To find the approximate weight of the human brain based on the circumference of the head using the given function \( w = \frac{c^3}{110 - 1500c} \), we can compute it for \( c = 30 \) cm and \( c = 40 \) cm.

### For \( c = 30 \) cm:
\[
w = \frac{30^3}{110 - 1500 \cdot 30}
\]
Calculating \( 30^3 \):
\[
30^3 = 27000
\]
Now, calculating \( 110 - 1500 \cdot 30 \):
\[
1500 \cdot 30 = 45000
\]
\[
110 - 45000 = -44990
\]
Now substituting these values back into the weight function:
\[
w = \frac{27000}{-44990} \approx -0.6001 \text{ grams}
\]

Since the weight cannot be negative, this implies that a circumference of 30 cm results in an undefined or negative weight, which suggests that the formula does not yield a realistic value for this circumference.

### For \( c = 40 \) cm:
\[
w = \frac{40^3}{110 - 1500 \cdot 40}
\]
Calculating \( 40^3 \):
\[
40^3 = 64000
\]
Now, calculating \( 110 - 1500 \cdot 40 \):
\[
1500 \cdot 40 = 60000
\]
\[
110 - 60000 = -59990
\]
Substituting these values back into the weight function:
\[
w = \frac{64000}{-59990} \approx -1.0667 \text{ grams}
\]

Again, the weight is negative, indicating that the formula is not applicable for a circumference of 40 cm either.

### Summary:

- For \( c = 30 \) cm, the weight \( w \) is approximately \(-0.6001\) grams (not realistic).
- For \( c = 40 \) cm, the weight \( w \) is approximately \(-1.0667\) grams (not realistic).

The formula appears to have limitations in its domain that prevent it from providing a valid output for the specified circumference values.