Chapter 3.2, Problem 38E

To determine

a.

To graph:

The graph of the weight gain and loss during pregnancy and the $20$ weeks following the birth of the baby by assuming that the pregnancy lasts $40$ weeks and the delivery occurs immediately after this interval.

Solution:

The graph of the weight gain and loss during pregnancy is as follows:

Explanation:

Given information:

A women’s weight naturally increases during pregnancy. When she delivers, her weight decreases immediately by the weight of the child. Suppose a $120\text{lb}$ woman gains $27\text{lb}$ during pregnancy, delivers a $7\text{lb}$ baby, and then, through diet and exercise, loses the remaining weight during the next $20$ weeks. The weight gain or loss before and after birth is linear.

Formula used:

(1) Equation of the line passing through two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is $\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$

Calculation:

Here, the vertical axis represents the weight and the horizontal axis represents the time period.

At time $t=0$, that is, at the start of pregnancy, weight is $120\text{lb}$, which increased after 40 weeks by $27\text{lb}$, so at $t=40$, weight is $120+27=147\text{lb}$. Immediately after delivery, that is, after $40$ weeks, the weight of the woman decreases by $7\text{lb}$, so at $t=40.1$, weight is $140\text{lb}$, and remaining weight loss occurs during the next $20$ weeks, so at $40+20=60$ weeks, weight is $120\text{lb}$.

So, coordinates are $\left(0,120\right),\left(40,147\right),\left(40.1,140\right)$, and $\left(60,120\right)$.

Now to find the slope of the line passing through the coordinates $\left(0,120\right)$ and $\left(40,147\right)$.

By using formula for the equation of the line passing through two points, it gives

$\frac{w-120}{t-0}=\frac{147-120}{40-0}$

By simplifying,

$\Rightarrow \frac{w-120}{t}=\frac{27}{40}$

By multiplying by $t$ on both sides,

$\Rightarrow w-120=\frac{27}{40}t$

By adding $120$ on both sides,

$\Rightarrow w=\frac{27}{40}t+120$

Thus, the equation of line is $w=\frac{27}{40}t+120$ for $0\le t\le 40$.

Now find the equation of the line passing through coordinates $\left(40.1,140\right)$ and $\left(60,120\right)$.

By using the formula for the equation of a line passing through two points, it gives

$\frac{w-120}{t-60}=\frac{140-120}{40.1-60}$

By simplifying,

$\Rightarrow \frac{w-120}{t-60}=\frac{20}{-19.9}$

By multiplying by $t-60$ on both sides,

$\Rightarrow w-120=-\frac{20}{19.9}\left(t-60\right)$

$\Rightarrow w-120=-\frac{20}{19.9}t+\frac{1200}{19.9}$

By adding $120$ on both sides,

$\Rightarrow w=-\frac{20}{19.9}t+\frac{1200}{19.9}+120$

$\Rightarrow w=-\frac{20}{19.9}t+\frac{1200+2388}{19.9}$

$\Rightarrow w=-\frac{20}{19.9}t+\frac{3588}{19.9}$

Thus, the equation of the line is $w=-\frac{20}{19.9}t+\frac{3588}{19.9}$ for $40<t\le 60$

Thus, we get the function for weight gain and loss during pregnancy and the $20$ weeks following the birth of the baby as $w=\{\begin{array}{c}\frac{27}{40}t+120,0\le t\le 40\\ -\frac{20}{19.9}t+\frac{3588}{19.9},40<t\le 60\end{array}$

Therefore, the graph of the weight gain and loss during pregnancy and the $20$ weeks following the birth of the baby is as follows:

To determine

b.