Fees Suppose you are going on vacation and leaving your dog...

Fees Suppose you are going on vacation and leaving your dog in a kennel.The Bowowery charges $\$ 5$ per day, which includes a one-time grooming treatment. The Poochpad charges $\$ 20$ per day and a one-time fee of $\$ 30$ for grooming. a. Write a system of equations to represent the cost $c$ for $d$ days that your dog will stay at a kennel. b. Using a graphing calculator, find the number of days for which the costs are the same. c. If your vacation is a week long, which kennel should you choose? Explain.

Fees Suppose you are going on vacation and leaving your dog in a kennel.The Bowowery charges $\$ 5$ per day, which includes a one-time grooming treatment. The Poochpad charges $\$ 20$ per day and a one-time fee of $\$ 30$ for grooming.
a. Write a system of equations to represent the cost $c$ for $d$ days that your dog will stay at a kennel.
b. Using a graphing calculator, find the number of days for which the costs are the same.
c. If your vacation is a week long, which kennel should you choose? Explain.

Key Concepts

Break-even Analysis

Break-even analysis identifies the point at which two options incur identical costs, allowing for a direct comparison of alternatives. This concept is crucial in decision making as it indicates the conditions under which two pricing models become financially equivalent.

System of Linear Equations

This concept involves setting up two or more linear equations that share common variables, with the goal of finding the values of these variables that satisfy every equation simultaneously. It is fundamental in algebra for modeling and solving real-world problems where multiple relationships or constraints coexist.

Linear Functions

Linear functions express relationships with a constant rate of change and are typically written in the form y = mx + b, where m is the slope and b is the y-intercept. They are used to model situations where outcomes vary in direct proportion to another variable, such as cost per day.

Graphical Analysis

Graphical analysis involves representing linear functions as lines on a coordinate graph to visualize their behavior. By plotting the equations, one can determine the point of intersection, which corresponds to the solution of a system of equations and helps interpret when two different conditions yield the same result.

Transcript

00:01 So we're given a word problem of sorts.

00:02 We're asked that we're going on vacation and we want to leave our dog in a kennel.

00:07 So the bowowery charges $5, excuse me, charges $25 per day, which includes a one -time grooming treatment.

00:17 Now the pooch pad charges $20 per day and a one -time fee of $30 for grooming.

00:23 So first we're asked to write a system of equations to represent the cost c for d days that your dog.

00:31 Dog will stay in that kennel.

00:33 So we're going to go ahead and do that here.

00:35 So our cost is c, so that's going to be kind of our y, right? so c is equal to, well, for the first one, we're going to do the bowery, right? so the bowowery, so that's going to be this one, and this would be the for the pooch pad.

00:50 So for the bauwowery, it's $25 per day.

00:54 So that's simply 25 times d, which is the day.

00:58 And for the pooch pad, that's $20 per day.

01:02 Plus a $30 flat rate, which is just plus 30.

01:05 So that's going to be our two equations for the cost.

01:09 So the next thing we're asked to do is using a graph and calculator, find the number of days for which the costs are the same.

01:17 So to do that, we're basically just going to set the equations equal to each other...

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