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Chapter 3: Problem 7

A premature baby needs 6 grams ( \(g\) ) of protein per day. The baby receives \(0.25 \mathrm{~g}\) of protein per hour. a. Write an equation that represents the relationship of the amount of protein the baby still needs on a particular day, \(y\), and the number of hours that have passed on that day, \(x\). b. Use the equation to find the amount of protein the baby still needs after \(8 \mathrm{hr}\) have passed. c. Find the \(x\)-intercept. d. Describe what the \(x\)-coordinate of the \(x\)-intercept represents.

Short Answer

Expert verified
Equation: \( y = 6 - 0.25x \). After 8 hours, 4 grams are still needed. The x-intercept is 24, meaning the baby will have received all the required protein after 24 hours.

Step by step solution

01

Identify the total protein needed

The total amount of protein the baby needs per day is 6 grams.
02

Identify the rate of protein intake

The baby receives 0.25 grams of protein per hour.
03

Write the equation

The amount of protein the baby still needs, denoted by y, can be represented by the equation: \[ y = 6 - 0.25x \] where x is the number of hours that have passed.
04

Calculate the amount of protein needed after 8 hours

Substitute \( x = 8 \) into the equation: \[ y = 6 - 0.25(8) \] \[ y = 6 - 2 \] \[ y = 4 \] After 8 hours, the baby still needs 4 grams of protein.
05

Find the x-intercept

The x-intercept is where y = 0. Set y to 0 in the equation and solve for x: \[ 0 = 6 - 0.25x \] \[ 0.25x = 6 \] \[ x = \frac{6}{0.25} \] \[ x = 24 \] The x-intercept is at (24, 0).
06

Describe the x-intercept

The x-coordinate of the x-intercept represents the number of hours needed for the baby to receive the entire 6 grams of protein required. In this case, it is 24 hours.

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Key Concepts

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

Translating Word Problems into Equations
Word problems can seem daunting at first, but breaking them down into smaller parts helps. The first step is to identify the quantities involved and their relationships. In our exercise, the baby needs 6 grams of protein per day. The baby receives protein at a continuous rate of 0.25 grams per hour. To translate this into an equation:
  • Let y represent the remaining amount of protein needed by the baby.
  • Let x represent the number of hours that have passed.
The relationship can be expressed as: \( y = 6 - 0.25x \). This equation shows that after every hour, the baby's remaining protein requirement decreases by 0.25 grams. By translating the given information into an equation, we make it easier to analyze and solve the problem.
Linear Equations
A linear equation is one that forms a straight line when graphed. It typically looks like \( y = mx + b \), where:
  • m is the slope (the rate of change)
  • b is the y-intercept (the starting value when x is 0)
In our equation \( y = 6 - 0.25x \), m is -0.25, showing that for every additional hour, the required amount of protein decreases by 0.25 grams. The b value of 6 represents the starting protein requirement for the day. Linear equations like this are straightforward to work with given their simplicity and predictability.
Solving for Variables
Solving for variables entails finding the value of the unknowns. For example, to find out how much protein the baby needs after 8 hours, substitute x = 8 into the equation: \[ y = 6 - 0.25(8) \] \[ y = 6 - 2 \] \[ y = 4 \] This means after 8 hours, the baby still needs 4 grams of protein. To find the x-intercept, set y to 0 and solve for x: \[ 0 = 6 - 0.25x \] \[ 0.25x = 6 \] \[ x = \frac{6}{0.25} \] \[ x = 24 \] Thus, the baby will receive the full 6 grams of protein after 24 hours. The x-intercept gives a crucial insight into when a specific condition is met鈥攊n this case, the complete daily protein requirement.

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Most popular questions from this chapter

For exercises 91-94, use a graphing calculator to graph each equation. Choose a window that shows the \(x\)-intercept and \(y\)-intercept. Sketch the graph; describe the window. \(y=-2 x+5\)

For exercises 97-98, some students find it helpful to use their learning preferences as a guide in how to study. Visual Learner \- Take detailed notes during class. Use colored pens and highlighters. \- Reorganize and rewrite notes after class; draw diagrams that summarize what you have learned. \- Read your book; watch the videos or DVDs for this text. \- Use flash cards for memory work. \- Sit where you can see everything in the classroom. Turn your phone or tablet off so that you are not distracted. Auditory Learner \- With permission, record your class. Take only brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Restate the main ideas aloud to yourself. Use videos and DVDs to fill in anything you missed in class. \- Talk to yourself as you do your homework. Explain each step to yourself. \- Do memory work by repeating definitions aloud. Listen to a recording of the words and definitions. Create songs that help you remember a definition. \- Sit where you can hear everything. Turn your phone or tablet off so that you are not distracted. Kinesthetic Learner \- With permission, record your class. Take brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Draw pictures. Use videos and DVDs to fill in anything you missed during class. -With your finger, trace diagrams and graphs. Do not just look at them. \- Imagine symbols such as variables as three-dimensional objects or even cartoon characters. Imagine yourself counting them, combining them, or subtracting them. \- Do memory work as you exercise or walk to your car. Walk around your room as you repeat definitions. You may find it helpful to come up with physical motions and/or a song that correspond to a procedure. \- If your class is mostly lecture, prepare yourself mentally before you walk into class to concentrate and not daydream. Turn your phone or tablet off so that you are not distracted. Identify any of the strategies listed above that you currently use to study math.

Use the slope formula to find the slope of the line that passes through the points. \((0,-4) ;(-3,0)\)

Explain why the slope of a vertical line is undefined.

Use the slope formula to find the slope of the line that passes through the points. \(\left(0, \frac{4}{5}\right) ;\left(\frac{1}{5}, 0\right)\)
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