Health A weight-loss counselor has prepared a program...

$$\begin{array}{c}{\text { Health } \quad \mathrm{A} \text { weight-loss counselor has prepared a program of }} \\ {\text { diet and exercise for a client. If the client sticks to the program, }} \\ {\text { the weight loss that can be expected (in pounds per week) is }} \\ {\text { given by }} \\ {\text { Weight loss }=f(n, c)=\frac{1}{8} n^{2}-\frac{1}{5} c+\frac{1937}{8}}\end{array}$$ where $c$ is the average daily calorie intake for the week and $n$ is the number of 40 -minute acrobic workouts per week. (a) How many pounds can the client expect to lose by eating an average of 1200 cal per day and participating in four 40 -minute workouts in a week? (b) Find and interpret af/\partialn. (c) The client currently averages 1100 cal per day and does three 40 -minute workouts each week. What would be the approximate impact on weekly weight loss of adding a fourth workout per week?

$$\begin{array}{c}{\text { Health } \quad \mathrm{A} \text { weight-loss counselor has prepared a program of }} \\ {\text { diet and exercise for a client. If the client sticks to the program, }} \\ {\text { the weight loss that can be expected (in pounds per week) is }} \\ {\text { given by }} \\ {\text { Weight loss }=f(n, c)=\frac{1}{8} n^{2}-\frac{1}{5} c+\frac{1937}{8}}\end{array}$$
where $c$ is the average daily calorie intake for the week and $n$ is
the number of 40 -minute acrobic workouts per week.
(a) How many pounds can the client expect to lose by eating
an average of 1200 cal per day and participating in four
40 -minute workouts in a week?
(b) Find and interpret af/\partialn.
(c) The client currently averages 1100 cal per day and does
three 40 -minute workouts each week. What would be the
approximate impact on weekly weight loss of adding a
fourth workout per week?

Key Concepts

Multivariable Function

A multivariable function is a function that depends on multiple input variables. In the context of the problem, the weight loss is modeled as a function of two independent variables—number of workouts (n) and daily calorie intake (c)—which allows for analyzing how changes in these inputs can affect the weight loss outcome.

Function Evaluation

Function evaluation involves substituting specific values for the variables in the function to compute a precise numerical output. It is used here to calculate expected weekly weight loss by plugging given numbers for workouts and calorie intake into the weight loss function.

Partial Derivative

A partial derivative represents the rate of change of a multivariable function with respect to one variable while keeping the other variables constant. In this problem, the partial derivative with respect to n reveals how weight loss is expected to change with an additional workout session when calorie intake remains unchanged.

Linear Approximation

Linear approximation uses the derivative(s) of a function to estimate small changes in the output due to small changes in the input variables. It is applied here to approximate the impact on weight loss when making a minor adjustment, such as adding one more workout per week.

Transcript

00:01 The question is asking the following.

00:03 A weight loss counselor has prepared a program of diet and exercise for a client.

00:07 If the client sticks to the program, the weight loss can be expected to be a function of n and c to be equal to 1 over 8 n squared minus 1 over 5c plus 1937 over 8, where this n is the number of 40 minutes workouts per week and the c is the average daily calorie intake for the week.

00:27 The first part of the question is asking how many pounds can the client expect to lose by eating an average of 1 ,200 calories per day and participating in four 40 -minute workouts in a week.

00:41 So i am given that c is equal to 1 ,200 and n is equal to 4, and i'm asked what is the weight loss that i should expect if i eat this much and if i exercise this much? so i have a function f of n and c.

00:58 I'm just going to use the values of n and c that i'm given in the problem and plug them into that function and calculate the expected weight loss.

01:06 So f of 4 and 1 ,200 is going to be 1 over 8 times 4 squared because n is equal to 4, minus 1 over 5c, which is 1 ,200, plus 1937 over 8.

01:28 This is equal to 1 over 8 times 4 squared which is 2 minus 1 over 5 times 1 ,200 this is equal to 240 and 1937 over 8 is equal to 242 .125 those values i just got them by plugging them into the calculator doing this simple addition i'm going to get 4 .125 pounds so to summarize if i if the client eats 1 ,200 calories per day and exercises for four for and he has four 40 minute workouts then he should expect to lose around four four point one to five pounds the second part of the question is asking what is partial f by partial n and to interpret it so we have our function we want to get partial f by partial n so this means i'm going to have to differentiate my function f with respect to n and i'm going to treat c as a constant so the first part of the problem which is 1 over 8 n squared this has an n in it so its derivative will be 1 over 8 times 2n and then the remaining part of the problem does not have any ends is just a constant so the derivative of a constant with respect to the variable n is just 0 so my partial f partial n will be equal to n over 4 and what does this mean this partial f by partial n this is the rate of change in unit loss for a unit change in the number of 40 minute workouts given that i'm holding c as a constant so this basically means how much do i expect the weight loss to change if i change the number of workouts by one unit when assuming that c is held constant and does not change the last part of the problem is asking the client it says the client currently averages 1 ,100 calories per day and does 3 40 -minute workouts each week.

03:37 What would be the approximate impact on weekly weight loss of adding a fourth workout per week? so in this problem, the customer, he's eating calories of 1 ,100, and n is equal to 3.

03:53 And they're asking us, what will be, what is he, what would be the approximate impact on his weight loss by adding a fourth workout in the week...

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