91Ó°ÊÓ

Between the ages of 7 and 11 years, the weight \(w\), in pounds, of a certain girl is given by the formula $$ w=8 t $$ Here \(t\) represents her age in years. a. Use a formula to express the age \(t\) of the girl as a function of her weight \(w\). b. At what age does she attain a weight of 68 pounds? c. The height \(h\), in inches, of this girl during the same period is given by the formula $$ h=1.8 t+40 . $$ i. Use your answer to part b to determine how tall she is when she weighs 68 pounds. ii. Use a formula to express the height \(h\) of the girl as a function of her weight \(w\). iii. Answer the question in part \(i\) again, this time using your answer to part ii. c. The height \(h\), in inches, of this girl during the same period is given by the formula $$ h=1.8 t+40 . $$ i. Use your answer to part b to determine how tall she is when she weighs 68 pounds. ii. Use a formula to express the height \(h\) of the girl as a function of her weight \(w\). iii. Answer the question in part \(i\) again, this time using your answer to part ii.

Step by step solution

01

Expressing Age as a Function of Weight

The original formula is \( w = 8t \). To express the age \( t \) as a function of weight \( w \), we need to solve for \( t \). Divide both sides of the equation by 8:\[ t = \frac{w}{8} \].So, the age \( t \) is expressed as \( t = \frac{w}{8} \) when the weight is \( w \).

02

Determine Age for 68 Pounds

Using the formula \( t = \frac{w}{8} \), substitute \( w = 68 \):\[ t = \frac{68}{8} \].Calculating this gives \( t = 8.5 \). Thus, she is 8.5 years old when she weighs 68 pounds.

03

Determine Height at 68 Pounds

Using \( t = 8.5 \), substitute this into the height formula \( h = 1.8t + 40 \):\[ h = 1.8(8.5) + 40 \].Calculate \( h = 15.3 + 40 \), so \( h = 55.3 \) inches. Thus, she is 55.3 inches tall at 68 pounds.

04

Express Height as a Function of Weight

We need to express height \( h \) in terms of weight \( w \). First recall \( h = 1.8t + 40 \) and substitute \( t = \frac{w}{8} \) from Step 1:\[ h = 1.8 \left(\frac{w}{8}\right) + 40 \].Simplify this to get\[ h = \frac{1.8w}{8} + 40 \] or equivalently \[ h = 0.225w + 40 \].

05

Re-calculate Height Using New Formula

Using the new formula for height as a function of weight \( h = 0.225w + 40 \), substitute \( w = 68 \):\[ h = 0.225(68) + 40 \].Calculate \( h = 15.3 + 40 \), which gives \( h = 55.3 \) inches. This result confirms the previous height calculation from Step 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Weight-Age Relationship

Understanding the relationship between weight and age can help us predict how a person's weight might change as they grow older. In the exercise provided, we see that the weight \( w \) of the girl is described with the formula \( w = 8t \). Here, \( t \) represents her age in years, and this suggests that her weight increases by 8 pounds for every year of age. This kind of relationship where the weight varies proportionally with age is quite common during childhood growth phases. To express age as a function of weight, we rearrange the equation. Solving for \( t \), or the girl's age, we find that \( t = \frac{w}{8} \). This means if you know the girl's weight, you can calculate her age by dividing her weight by 8. For instance, if she weighs 68 pounds, substituting this weight into our age function gives us \( t = \frac{68}{8} = 8.5 \). Thus, she is 8.5 years old when she reaches this weight.

Linear Equations

Linear equations are mathematical statements that describe a straight-line relationship between two variables. In this context, both the weight-age relationship and the height-weight relationship are modeled using linear equations. For example, the equation \( w = 8t \) describes how weight changes linearly with age.The main feature of linear equations is their predictability; they allow us to easily calculate one variable if we know the other. For instance, when given the weight, we can determine the age using the rearranged equation \( t = \frac{w}{8} \). This formula decides how the variables interact - a core idea behind algebraic functions.Understanding these relationships helps us communicate complex ideas in simple terms, making these problems much easier to manage. Whether it's about identifying a point in time when the girl reaches a particular weight or height, linear equations offer a straightforward technique to find the required information.

Height-Weight Relationship

Exploring the connection between height and weight in the provided exercise reveals another linear relationship. The height \( h \) of the girl is given by the equation \( h = 1.8t + 40 \). Initially, this formula describes height as a function of age \( t \), indicating that the height increases by 1.8 inches with each additional year of age, plus a base height of 40 inches.To represent height as a direct function of weight, we substitute \( t = \frac{w}{8} \) (from weight-age relationship) into the height equation. This results in the new formula \( h = 0.225w + 40 \), articulating height directly as a function of weight. To find how tall she is at a specific weight, like 68 pounds, we substitute \( w = 68 \) into this equation:

• \( h = 0.225 \times 68 + 40 \)
• \( h = 15.3 + 40 \)

This gives a height of 55.3 inches, verifying our earlier solution. By understanding these linear relationships, we can confidently determine height from weight using this straightforward approach.