Have you ever been to a Waterpark? What’s your favorite waterpark? What’s your favorite waterslide? Let me introduce you to the tallest waterslide in the world…

Located in Fortaleza, Brazil Built in stories high Riders reach a speed of 65mph Ride lasts 5 seconds Listed in the Guinness Book of World Records One rider commented, “When you slide down, you basically fly because the fall is too steep so you’re in mid-air for most of it.” Experience it yourself! Insano: Tallest waterslide in the world

Suppose Aly and Dwayne work at the Beach Park water park in Brazil. They each are responsible for draining the pool at the bottom of their ride. Their pools are different sizes and therefore hold different amounts of water. If they each use a pump to remove the water, how can we use math to figure out after how many minutes Aly and Dwayne’s pools will have the same amount of water in them? Today we’re going to use math to figure out the following scenario…

Lesson 3. 5 Identifying Solutions Concept: Identifying Solutions EQ: How do we identify and interpret the solutions of an equation f(x) = g(x)? Standard: REI Vocabulary: Expenses, Income, Profit, Break-even point

Let’s Review A solution to a system of equations is a value that makes both equations true. y=-x -4 y=2x -1 (-1,-3) -3=-(-1) -4 -3=-3 ✓ -3=2(-1) -1 -3= = -3 ✓

Let’s Review The point where two lines intersect is a solution to both equations. In real world problems, we are often only concerned with the x-coordinate.

Let’s Review Remember that in real-world problems, the slope of the equation is the amount that describes the rate of change, and the y-intercept is the amount that represents the initial value. For business problems that deal with making a profit, the break-even point is when the expenses and the income are equal. In other words you don’t make money nor lose money…your profit is $0.

Let’s Review Words to know for any business problems:  Expenses - the money spent to purchase your product or equipment  Income - the total money obtained from selling your product.  Profit - the expenses subtracted from the income.  Break-even point - the point where the expenses and the income are equal. In other words you don’t make money nor lose money…your profit is $0.

In this lesson you will learn to find the x-coordinate of the intersection of two linear functions in three different ways: 1. By observing their graphs 2. Making a table 3. Setting the functions equal to each other (algebraically)

Core Lesson Aly and Dwayne work at a water park and have to drain the water from the small pool at the bottom of their ride at the end of the month. Each uses a pump to remove the water. Aly’s pool has 35,000 gallons of water in it and drains at a rate of 1,750 gallons a minute. Dwayne’s pool has 30,000 gallons of water in it and drains at a rate of 1,000 gallons a minute. After approximately how many minutes will Aly and Dwayne’s pools have the same amount of water in them? Example 1

Core Lesson We need to write 2 equations! Aly’s pool has 35,000 gallons of water in it and drains at a rate of 1,750 gallons a minute. Dwayne’s pool has 30,000 gallons of water in it and drains at a rate of 1,000 gallons a minute. First we can identify our slope and y-intercept. slope y-intercept Aly’s a(x)= -1,750x + 35,000Dwayne’s d(x)= -1,000x + 30,000 Example 1 x=# of minutes; a(x) & d(x)=amount of water left in pool Both of the slopes will be negative because the water is leaving the pools.

Core Lesson The graph below represents the amount of water in Aly’s pool, a(x), and Dwayne’s pool, d(x), over time. After how many minutes will Aly’s pool and Dwayne’s pool have the same amount of water? Aly’s pool Dwayne’s pool Find the point of intersection. Approximate the x-coordinate. Aly’s pool and Dwayne’s pool will have an equal amount of water after 10 minutes. In a problem like this, we are only concerned with the x-coordinate. Example 1

Core Lesson Aly’s Pool Dwayne’s Pool Here, the graph helps us solve, but graphing can also help us to estimate the solution. Can you think of a problem where an approximation might be sufficient? Example 1

Core Lesson A large cheese pizza at Paradise Pizzeria costs $6.80 plus $0.90 for each topping. The cost of a large cheese pizza at Geno’s Pizza is $7.30 plus $0.65 for each topping. How many toppings need to be added to a large cheese pizza from Paradise and Geno’s in order for the pizzas to cost the same, not including tax? First we can identify our slope and y-intercept. slope y-intercept We need to write 2 equations! Paradise p(x)=.90x Geno’s g(x)=.65x Example 2

Core Lesson Geno’s g(x)=.65x Paradise p(x)=.90x x=# of toppings; p(x) & g(x)=total cost The pizzas cost the same! After adding two toppings, the pizzas will cost the same! g(x)=.65x (0) = (1) = (2) = 8.60 x p(x)=.90x (0) = ( = (2) = 8.60 Example 2 Now we make a chart to organize our data! We need one chart but 3 columns for two equations!

Core Lesson Eric sells model cars from a booth at a local flea market. He purchases each model car from a distributor for $12, and the flea market charges him a booth fee of $50. Eric sells each model car for $20. How many model cars must Eric sell in order to reach the break-even point? Example 3

Core Lesson Eric sells model cars from a booth at a local flea market. He purchases each model car from a distributor for $12, and the flea market charges him a booth fee of $50. Eric sells each model car for $20. First we can identify our slope and y-intercept. slope y-intercept slope We need to write 2 equations! e(x)= 12x + 50 Example 3 x=# of model cars; e(x)=Eric’s expenses; f(x)= Eric’s Income f(x)= 20x

Core Lesson Since both e(x) and f(x) are are equal to “y”, you can set the equations equal to each other and solve for “x”. Eric’s Expenses e(x)=12x + 50Eric’s Income f(x)=20x e(x) = f(x) 12x + 50 = 20x 50 = 8x 6.25 = x Eric needs to sell more than 6 model cars to break even! Example 3

Core Lesson Profit = Income – Expenses So take the two given functions and subtract them. Eric’s Expenses e(x)=12x + 50 Example 3 How can we write a function to represent Eric’s Profit? Eric’s Income f(x)=20x P(x) = f(x) – e(x) P(x) = 20x – (12x + 50) P(x) = 8x - 50

Core Lesson You Try 1 – Solve using graphing Chen starts his own lawn mowing business. He initially spends $180 on a new lawnmower. For each yard he mows, he receives $20 and spends $4 on gas. If x represents the # of lawns, then let Chen’s expenses be modeled by the function m(x)=4x and his income be modeled by the function p(x) = 20x How many lawns must Chen mow to break-even?

Core Lesson Text Away cell phone company charges a flat rate of $30 per month plus $0.20 per text. It’s Your Dime cell phone company charges a flat rate of $20 per month plus $0.40 per text. If x represents the # of texts, then let your Text Away bill be modeled by the function t(x)=.20x + 30 and Your Dime bill be modeled by the function d(x) =.40x + 20 How many texts must you send before your bill for each company will be the same? You Try 3 – Solve using algebra

Core Lesson You Try 2 – Solve using a table Olivia is building birdhouses to raise money for a trip to Hawaii. She spends a total of $30 on the tools needed to build the houses. The material to build each birdhouse costs $3.25. Olivia sells each birdhouse for $10. If x represents the # of birdhouses, then let Olivia’s expenses be modeled by the function b(x)=3.25x + 30 and her income be modeled by the function p(x) = 10x How many birdhouses must Olivia sell to break-even?

Suppose a friend of yours in this class was absent today and missed this lesson… They send you a text message later asking you what they missed and how do they do the homework. Write me your response to them. Message to Absent Student