Q19. - Algebra 1

Question

The number of pints of blood in a human body varies directly with the person’s weight. A person who weighs 120 pounds has about 8.4 pints of blood in his or her body.

 

a. Write and graph an equation relating weight and amount of blood in a person’s body.

b. Predict the weight of a person whose body holds 12 pints of blood.

Step-by-Step Solution

Answer

a. The equation relating weight and amount of blood in a person’s body is $p=\frac{84}{100}w.$

b. The weight of a person whose body holds 12 pints of blood is 171.43 pounds.

Part a. Step 1. State the concept of Direct Proportionality.

Direct proportion is a mathematical comparison between two numbers where the ratio of the two numbers is equal to a constant value. The proportion definition says that when two ratios are equivalent, they are in proportion. The symbol used to relate the proportions is "$\propto$".

 

Here number of pints (p) of blood in a human body varies directly with the person weight (w).

 That is, $p\propto w$

 Therefore $p=kw\dots \left(1\right)$

Where, k is a constant.

Part a. Step 2. Find the value of ‘ k ’.

Find the value of the constant ‘k’ by substituting the value $p=8.4$ when $w=120$

In equation (1)

$$\begin{gathered} 8.4=k\left(120\right) \\ \frac{8.4}{120}=k \\ \frac{84}{1200}=k \\ k=\frac{84}{1200} \end{gathered}$$

Part a. Step 3. Find the required equation.

Substitute the value of k in equation (1) to get the required equation.

$p=\frac{84}{1200}w\dots \left(2\right)$

 

This is the required equation.

Part b. Step 1. State the concept of Direct Proportionality.

Direct proportion is a mathematical comparison between two numbers where the ratio of the two numbers is equal to a constant value. The proportion definition says that when two ratios are equivalent, they are in proportion. The symbol used to relate the proportions is "$\propto$".

 

Here number of pints (p) of blood in a human body varies directly with the person weight (w).

 

That is, $p\propto w$

Therefore $p=kw\dots \left(1\right)$

Where, k is a constant.

Part b. Step 2. Find the value of ‘ k ’ and the required equation.

First find the value of the constant ‘k’ by substituting the value $p=8.4$ when $w=120$ in $p=kw$

$$\begin{gathered} 8.4=k\left(120\right) \\ \frac{8.4}{120}=k \\ \frac{84}{1200}=k \\ k=\frac{84}{1200} \end{gathered}$$

Substitute the value of k in equation (1) 

$p=\frac{84}{1200}w\dots \left(2\right)$ 

This is the required equation.

Part b. Step 3. Find the weight of the person when p = 12    .

Substitute $p=12$ in equation (2) to get the weight of the person.

$$\begin{gathered} 12=\frac{84}{1200}w \\ 12\times 1200=\frac{84}{1200}\times 1200w \\ 14400=84w \\ \frac{14400}{84}=\frac{84}{84}w \\ 171.43=w \\ w=171.43 \end{gathered}$$

Thus, the weight of the person is 171.43 pounds.