/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 82 For exercises \(67-82\), use the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 82

For exercises \(67-82\), use the five steps and a proportion. Cyclosporine is an anti-rejection drug given to organ transplant patients. A bottle contains \(50 \mathrm{~mL}\) of liquid. Each milliliter of liquid contains \(100 \mathrm{mg}\) of cyclosporine. A kidney transplant patient needs to take \(850 \mathrm{mg}\) of cyclosporine each day. Find the amount of solution that the patient should take each day.

Short Answer

Expert verified
The patient should take 8.5 mL of the solution each day.

Step by step solution

01

- Determine the known values

Identify the values given in the problem. The bottle contains 50 mL of liquid. Each mL of liquid contains 100 mg of cyclosporine. The patient needs 850 mg of cyclosporine each day.
02

- Set up the proportion

We need to determine how many mL the patient should take to get 850 mg. Set up the proportion using the known values: \[\frac{100 \text{ mg}}{1 \text{ mL}} = \frac{850 \text{ mg}}{x \text{ mL}}\]
03

- Solve for x

Cross-multiply to solve for x: \[100x = 850\]
04

- Divide both sides

Divide both sides by 100 to isolate x: \[x = \frac{850}{100} = 8.5 \text{ mL}\]
05

- Verify the solution

Check the solution by calculating the amount of cyclosporine in 8.5 mL of liquid: \[8.5 \text{ mL} \times 100 \text{ mg/mL} = 850 \text{ mg}\]The solution is correct.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Proportions
Understanding proportions is crucial when solving many algebra problems, including this one about medication dosage. A proportion is a statement where two ratios or fractions are equal. In this exercise, we use the relationship between milliliters (mL) and milligrams (mg) of cyclosporine to set up a proportion. This helps us find the amount of solution the patient needs. One key point is to ensure that both sides of the proportion are comparing the same type of units. Ratios, like \(\frac{100 \text{ mg}}{1 \text{ mL}}\), show the amount of cyclosporine per mL.
Cross-Multiplication
Cross-multiplication is a helpful technique to solve proportions. It involves multiplying across the equals sign diagonally. When you have a proportion \(\frac{a}{b} = \frac{c}{d}\), you can cross-multiply to find \(\text{a} \times \text{d} = \text{b} \times \text{c}\). In our exercise, we begin with the proportion \(\frac{100 \text{ mg}}{1 \text{ mL}} = \frac{850 \text{ mg}}{x \text{ mL}}\). By cross-multiplying, we get \(\text{100} \times \text{x} = 850 \text{ mg} \times 1 \text{ mL}\), leading us to the next steps of solving for \(\text{x}\).
Algebraic Equations
Once we've set up our proportion and cross-multiplied, we have an algebraic equation to solve. In this problem, solving \(\text{100x} = 850\) requires isolating \(\text{x}\). We do this by dividing both sides of the equation by 100: \[x = \frac{850}{100} = 8.5 \text{ mL}\]. Solving algebraic equations often involves these steps: simplifying expressions, isolating the variable, and performing arithmetic operations. Remember, the goal is to isolate the variable to find its value.
Unit Conversion
Unit conversion is essential in the context of proportions and real-world problems like medication dosages. Always keep track of units when solving problems. In this example, we convert from mg to mL using known ratios. Each mL contains 100 mg of cyclosporine, and the patient needs 850 mg. Our task is to convert 850 mg into the equivalent mL. By using the proportion \(\frac{100 \text{ mg}}{1 \text{ mL}} = \frac{850 \text{ mg}}{x \text{ mL}}\), we performed a unit conversion to find the correct dosage per day.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=3, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=9\).

For exercises \(45-48\), the formula \(R=\frac{U F}{P}\) describes the glomular filtration rate by a kidney \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ R \text { and } P \text { are constant; the relationship of } U \text { and } F \text {. } $$

Medical researchers collected data on 272 patients who were hospitalized for at least 24 hours with the 2009 H1N1 influenza in the United States from April 2009 to mid-June 2009. One out of four of these patients were admitted to an intensive care unit. About 9 out of 20 patients were children under the age of 18 years. Find the number of patients who were children. Round to the nearest whole number. (Source: www.nejm.org, Nov. 12, 2009)

A regulation basketball court in the NBA and the NCAA is \(94 \mathrm{ft}\) long and \(50 \mathrm{ft}\) wide. A regulation high school basketball court is \(84 \mathrm{ft}\) long and \(50 \mathrm{ft}\) wide. Find the percent increase in the area of an NCAA court compared to a high school court. Round to the nearest percent.

For exercises 49-52, the formula \(C=\frac{P_{m} P_{i}}{T F}\) describes the cost of insurance, \(C\). Is the relationship of the given variables a direct variation or an inverse variation? $$ P_{i}, T \text {, and } F \text { are constant; the relationship of } C \text { and } P_{m} \text {. } $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.