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Radical expressions and functions are commonly found in various fields of mathematics. This lesson will explore the use of radical functions and how they might be applied to a variety of everyday real-life situations, even just watching TV!

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

## How to Calculate the Speed of a Car?

After becoming a subscriber of a streaming platform, Magdalena became interested in a particular crime series. While watching an episode, she found out that after an accident, police can determine the speed of a car before the driver started braking.

The speed of a car can be estimated by using the following formula.
In this formula, represents the speed in miles per hour, is the coefficient of friction between the road surface and the tires, and the braking distance — the length of skid marks on the road — is measured in feet. Assume that In another episode, criminals organized illegal drag races. When crossing the finish line, the speed of a race car with a total weight of pounds can be modeled by the following function.
In this formula, is the power of the engine of the vehicle in horsepower and is measured in miles per hour. Because both radical functions measure speed and involve roots, Magdalena does not see any difference between them.
a Determine the domains and of the functions.
b Compare the average rate of change of the functions over the interval
c Graph the functions. What is their end behavior?

## Kleiber's Law

Magdalena shared her streaming account with Kriz so that they can watch a new nature documentary. The documentary is about the work of a Swiss agricultural biologist, Max Kleiber. He researched how the mass of an animal influences the animal's metabolic rate.

 Metabolic Rate The rate at which a person or an animal burns calories to maintain its body weight.
Kleiber's Law states that an animal's metabolic rate can be modeled by the following radical function.
In this function, is the mass of the animal in kilograms and is measured in kilocalories per day.
a Kriz has a couple of pets. Help Kriz calculate their and their pets' metabolic rates. Each mass is given in the table. Round to decimals if needed and write the metabolic rates in increasing order.
Mammal Mass
Mouse g
Kriz kg
Horse kg
b Consider two animals with weights and such that the first animal metabolizes times more calories per day than the second animal. How many times heavier than the second animal is the first animal?

### Hint

a Substitute each mass into the given formula. Use the Roots of Powers Property to evaluate the metabolic rate.
b Write the equation for the metabolic rate of a heavier animal. Substitute the formula stated in Kleiber's Law to solve for the weight.

### Solution

a The metabolic rate of Kriz and their animals can be calculated using the function stated in Kleiber's Law. Notice that not all masses are given in kilograms. The mass of a mouse can be written in kilograms by using the conversion factor
Now, substitute into to find the metabolic rate of a mouse.
Evaluate right-hand side
Next, Kriz's metabolic rate will be calculated. To do so, substitute into The Roots of Powers Property will be used to evaluate the rate.
Evaluate right-hand side

The same process can be used to calculate the metabolic rate of the horse.
Finally, the results can be summarized in a table.
Mammal Mass Metabolic Rate
Mouse
Kriz
Horse
b It is given that the metabolic rate of a heavier animal is times the metabolic rate
To determine the number of times the animal is heavier, the given formula for the metabolic rate will be substituted into the equation.
Now, solve the obtained equation for
Solve for
Evaluate right-hand side

This means that an animal that metabolizes times more calories per day is times heavier.

## A Ball Thrown Vertically

While watching a basketball game on the streaming platform, Kriz started thinking about the height of a thrown ball and how it relates to how much time passed since the throw. They once heard that the relationship is given by a quadratic function. However, Kriz wants to check whether this is also the case for a vertical throw. The following radical equation models how much time passed since the vertical throw.
In this formula, is the vertical velocity in feet per second, is the corresponding height of the ball, and the ball is thrown from a height of Both and are measured in feet.
a Help Kriz derive the formula for the height of the ball to check if it is modeled by a quadratic function.
b Assume that Kriz throws a ball from a height of feet with an initial vertical velocity of feet per second. What will the height of the ball be after seconds? Calculate the difference between the maximum and the obtained height.

### Hint

a Solve the given equation for Start by isolating the square root. Then, square the equation.
b Use the formula written in Part A. To evaluate the maximum height, find the coordinate of the vertex of the corresponding parabola.

### Solution

a The formula for the height of a ball can be derived by solving the given equation for To do so, start by isolating the square root.
Rewrite
Both sides of the equation can now be squared to solve for
Solve for
It has been calculated that This means that the height of the ball is indeed modeled by a quadratic function.
b To calculate the ball's height seconds after the throw, the formula derived in Part A will be used. It is given that the ball is thrown from a height of feet with an initial velocity of feet per second. Therefore, and will be substituted into the formula.
Now, substitute into the obtained quadratic equation to determine the desired height.
Evaluate right-hand side
After seconds, the ball will be at a height of feet. This is the height at which the ball was initially thrown. Now, the maximum height of the ball will be found. Notice that the function is already written in standard form.
To calculate the maximum value of the function, start by determining the coordinate of the vertex of the corresponding of parabola. The coordinate of the vertex is The maximum value can be calculated by substituting into the function rule.
Evaluate right-hand side
Finally, the difference between the maximum height and the height after seconds will be calculated.

## A Quarter Marathon

Magdalena and Kriz see a news story about the quarter marathon they ran last weekend. The distance of the marathon was kilometers. Participants were divided into groups starting at equal intervals. Magdalena and Kriz had different strategies for how to adjust their running pace to their physical condition. Magdalena's distance covered after minutes can be modeled by the following linear function.
In this function, is given in kilometers. On the other hand, Kriz's progress after minutes is modeled by a square root function.
Similarly, is measured in kilometers. For the following questions, assume that the marathon starts at
a What was the delay between Magdalena's and Kriz's start times?
b At what time(s) did Magdalena and Kriz run side by side? Solve the corresponding system of equations by graphing.

### Hint

a Determine at what values of Magdalena and Kriz start the marathon.
b The desired time is given by the values of such that the distances covered by Magdalena and Kriz are equal. Solve the equation by graphing.

### Solution

a It is given that the marathon started at Notice that when is greater than the distance covered by Magdalena was also greater than
This means that Magdalena started the race in the first group. In Kriz's case, the distance covered is given by a square root function. Since the distance cannot be an imaginary number, this function is defined for values of such that the radicand is non-negative.
Because the function is not defined for values of less than Kriz must not have started the race yet. Moreover, the square root of a positive number is also positive.
This means that Kriz started running minutes after Magdalena, at Consequently, the delay was minutes.
b Recall that Magdalena and Kriz ran in the same marathon. Since the delay between their starts was found in Part A, it can be assumed that both given functions share the same timeline Consider the functions modeling the distance each student ran one more time.
Any times at which Magdalena and Kriz ran side by side will be given by the values of such that their covered distances are equal.
The combined equation can be solved by graphing. Start with the linear function. To do so, the function rule must be written in slope-intercept form. Then, the slope and intercept can be easily identified.
Now, the obtained equation can be graphed by plotting its intercept. Then, the slope will be used to determine another point that satisfies the equation, and the points will be connected with a line. Next, the second equation will be graphed in the same coordinate plane by using a table of values. Remember to substitute so that the square root can be calculated right away.

All these points will be plotted and connected with a smooth curve. According to the graph, Magdalena ran the first kilometers in minutes, while Kriz ran the first kilometers in minutes. They met minutes from the start of the marathon. After this time, Kriz was in front of Magdalena for a while.
However, Kriz's rhythm was declining while Magdalena's remained constant. Then, minutes after the start of the race, Magdalena caught up with Kriz and from then on, she took the lead until the end of the marathon. Nevertheless, notice that both ran the marathon in minutes.
It can be concluded that the distances covered by Magdalena and Kriz were equal at and This means that they ran side by side minutes and minutes after the start of the marathon.

## Period of a Pendulum

A mathematical pendulum is an idealized model of a simple pendulum. It consists of an object suspended from a fixed point with a string. The string cannot be stretched. Consider a pendulum with a period of seconds. The period of a mathematical pendulum, the time it takes to complete one full cycle in seconds, can be approximated for small swings by the following square root function.
Here, is the length in of the pendulum in feet and is gravity acceleration in feet per second squared. The pendulum that Kriz built has a length of feet.
a Assume that is feet per second squared and Write the function for the period of Kriz's pendulum depending on the number of feet by which the pendulum is lengthened. Round the constant to decimals if needed.
b Graph the obtained function. Determine the domain and range of the function. Choose the correct answer.
c How does the value of influence the pendulum's period?

### Hint

a What is the length of Kriz's pendulum after it is lengthened by
b Make a table of values to graph the function. Keep in mind that a real square root is defined for a non-negative radicand.
c What is the end behavior of the obtained function?

### Solution

a It is given that Kriz's pendulum has a length of feet. Because the pendulum is about to be lengthened by the total length is given by the following sum.
To write the desired function, substitute for into the given function for the period of a pendulum.
Next, substitute and to simplify the function rule.

Simplify right-hand side

b Before graphing the function obtained in Part A, its domain will be determined. Recall that a real square root is defined only for non-negative radicands.
Therefore, the domain of the function is all real numbers greater than or equal to Next, make a table of values to graph the function. Substitute values of so that the square root can be easily calculated.

Next, the points from the table will be plotted on a coordinate plane and connected with a smooth curve. Since the square root of a greater number is also greater, the function will increase to infinity. Additionally, the square root of a non-negative number is also non-negative. Consequently, the range of the function consists of numbers greater than or equal to The obtained domain and range correspond to option C.

c Once again, consider the graph of the function drawn in Part B. It was determined that the range of this function consists of real numbers greater than or equal to Also, because the value of the function keeps increasing without bound, the end behavior of the function is up.
Moreover, the duration of the period of the pendulum increases more and more slowly as the value of increases. Note that this relationship goes both ways. This means that the longer the period is, the longer the pendulum that creates it needs to be.

## Profit of Software Start-ups

In addition to watching TV, Magdalena also loves programming and developing simple applications for personal use in her spare time. She has recently started thinking about doing a summer internship at one of the local start-up companies. The following radical function estimates the annual profit of the first company in millions of dollars.
In this function, is the number of years after The following graph represents the function that models the profit of the second start-up. a Choose transformations that are applied to graph the function starting from the graph of the parent function