High School Algebra II Unlocked (2016)

Lesson 7.3. Modeling Situations With Equations and Inequalities

REVIEW

“Percent” means “out of 100,” so to convert a given percent to a decimal, divide by 100, or move the decimal point two places to the left.

EQUATIONS AND INEQUALITIES IN ONE VARIABLE

A family went out to eat at a restaurant, where they used a coupon that gave them a 20% discount on food (but not drinks), before tax. They paid a total of $50.40, including 8% tax and an 18% tip (both percents based on the pre-tax total). If their bill included $12 for drinks, how much did they save by using the coupon?

You can also write
the total amount of
their discounted food
order as the full-price
amount minus 20%
of that amount:
f − 0.2f. This also
simplifies to 0.8f.

We must write an equation that models this situation. Although we want to solve for the family’s savings in the end, the basic unknown information is how much the food portion of their order cost. Let f = the price of the food they ordered. With a 20% discount on food, they only paid 80% of the food’s price, or 0.8f.

The family also ordered $12 in drinks, so the total price of their order was 0.8f + 12. They also paid 8% tax and 18% tip on this amount. We can write the total amount they paid as (0.8f + 12) + 0.08(0.8f + 12) + 0.18(0.8f + 12), which simplifies, using the distributive property, as (1 + 0.08 + 0.18)(0.8f + 12), or 1.26(0.8f + 12). Set this expression equal to the total amount the family paid.

1.26(0.8f + 12) = 50.4

We can now solve the equation for f.

Alternatively, you
could first distribute
through to get
1.008f + 15.12 = 50.4,
then subtract 15.12
from both sides to
get 1.008f = 35.28,
and finally divide
both sides by 1.008
to get f = 35.

With the original price of the family’s food order, we can now calculate how much they would have paid without the coupon. The total price of their food and drinks would be 35 + 12 = 47 dollars. They would still have paid an additional 8% tax and 18% tip, for a total extra 26% of the pre-tax total.

1.26(47) = 59.22

Without the coupon, the family would have paid $59.22 total. Subtract the amount they actually paid.

59.22 − 50.40 = 8.82

The family saved a total of $8.82 by using the coupon.

Alternatively, we could
calculate the savings
by finding 20% of the
total food price, $35,
and adding tax and tip
on that amount. 20%
of $35 is $7. The total
of $7, 8% of $7, and
18% of $7 is $8.82.

In Sam’s hometown, the population has been increasing at an annual rate of 2.5% since 2010, when the population was 790,390. If the population continues to increase at this rate, in what year will the population be greater than one million?

Because the increase in population is a percent of each previous year’s population, we must use an exponential expression to represent Sam’s hometown population at a given time. Let t represent years since 2010 (so t = 0 represents the year 2010). The city’s population is multiplied by 1.025 each year. (The 1 represents the same population number repeated, and the 0.025 represents the additional 2.5% of that number in growth.) So, the population in 2010 is multiplied by 1.025 t times to find the population after t years: 790,390 ⋅ 1.025^t.

In Chapter 6, Lesson
6.3, we learned that
real-life exponential
growth functions are
typically written in the
form f(x) = a ⋅ b^x, with
b > 1. Here, we must
write an exponential
expression to represent
the population
growth but then also
use the expression
in an inequality to
solve for when its
value is greater
than one million.

Let’s set up an inequality to find when this population is greater than one million (1,000,000).

Sam’s hometown will have a population of over 1,000,000 about 9.526 years after 2010. We must round up to the next whole-number year. The population will be greater than 1,000,000 in the year 2020.

EQUATIONS AND INEQUALITIES IN TWO VARIABLES

An equation or inequality in two variables models the relationship between two different sets of values. In many math problems, you are given a specific value for one of the variables and then must use the equation or inequality to solve for the other. Other times, you may need only to express the relationship as an equation or inequality, or as a system of equations or inequalities.

For a bike ride at a constant speed, on flat ground with no wind, Hiroshi found that his energy output for the ride was related to the speed at which he rode. For every additional mile per hour (mph) in his speed, he used an additional 20% calories per hour. When he rides at 10 mph for an entire bike ride, he burns 124 calories per hour. Write a function that relates the calories per hour, c(x), that Hiroshi burns on a bike ride, to his velocity, x, in miles per hour.

Hiroshi’s base daily caloric need is 2500 calories, for a day when he does not ride his bike. Write a function to represent Hiroshi’s total caloric needs, d(x), for a day in which he took a 2-hour bike ride at a speed of x miles per hour. Use this function to find the total calories he needs for a day in which he rides for 2 hours at 15 mph.

The first function relates calories burned per hour to speed in mph. For each increase of 1 mph, Hiroshi uses an additional 20% calories per hour. So, this is an exponential relationship, with some original value multiplied by 1.2 for each increase of 1 mph. The base is 1.2, and the exponent is x, the speed in mph. Use the given data to solve for the unknown original value, c₀. Write an exponential function with the original value, c₀, multiplied by 1.2^x to get c(x).

Similar to Example
11, we must set
up an exponential
function to represent
the situation, but in
this case we are not
given the coefficient
value. To solve for it,
we must use the given
pair of x and c(x)
values, (10, 124).

Let’s round c₀ off to 20. The function relating calories per hour, c(x), that Hiroshi burns on a bike ride, to his velocity, x, in mph, is c(x) = 20 ⋅ 1.2^x.

Now we want to create a function that relates the total number of calories needed one day to the speed at which Hiroshi rode for a 2-hour bike ride that day. We can use the function relating calories per hour to speed, to find the total calories used for a 2-hour bike ride. Multiply the calories per hour by the number of hours: 2 ⋅ (20 ⋅ 1.2^x), or 40 ⋅ 1.2^x. Additionally, Hiroshi has a base daily caloric need of 2500 calories. So, for a day in which he takes a 2-hour bike ride at a velocity of x mph, he needs a total of d(x) calories, given by the function d(x) = 40 ⋅ 1.2^x + 2500.

On a day in which Hiroshi rode for 2 hours at 15 mph, x = 15. Substitute this value into d(x) = 40 ⋅ 1.2^x + 2500.

Hiroshi needs about 3116 calories for this day.

Here is how you may see modeling with equations on the SAT.

An aquarium is putting in a new 180-gallon tank to display fish and plants that thrive in a salinity of 1.5 ppt (parts per thousand). An aquarium employee will mix seawater having a salinity of 35 ppt with freshwater having a salinity of 0.5 ppt. Solving which of the following systems of equations yields the number of gallons of seawater, s, and the number of gallons of freshwater, f, that will be combined to fill the tank?

A) s + f = 180
0.035s + 0.0005f = 0.27

B) s + f = 180
(0.035 + 0.0005)(s + f) = 0.0015

C) s + f = 180
0.035s + 0.0005f = 0.0015

D) s + f = 0.0015
0.035s + 0.0005f = 180