Lesson Using quadratic equations to solve word problems

Using quadratic equations to solve word problems

In this lesson we present some typical word problems that may be solved using quadratic equations.
Solution of quadratic equations is described in the lesson Introduction into Quadratic Equations in this module.

Problem 1. Motorboat moving upstream and downstream on a river

A motorboat makes a round trip on a river 56 miles upstream and 56 miles downstream, maintaining the constant speed 15 miles per hour relative to the water.
The entire trip up and back takes 7.5 hours.
What is the speed of the current?

Solution
Denote the unknown current speed of the river as miles/hour.
When motorboat moves upstream, its speed relative to the bank of the river is miles/hour, and the time spent moving upstream is hours.
When motorboat moves downstream, its speed relative to the bank of the river is miles/hour, and the time spent moving downstream is hours.
So, the total time up and back is , and it is equal to 7.5 hours, according to the problem input.
This gives an equation .
To simplify the equation, multiply both sides by and collect common terms. Step by step, you get
,
,
,
.

Answer. The speed of the current is 1 mile/hour.

For more examples of solved word problems of this type see the lesson Wind and Current problems solvable by quadratic equations
under the topic Travel and Distance in the section Word problems in this site.

Problem 2. Working together and separately to complete a job

Andrew and Bill, working together, can cover the roof of a house in 6 days.
Andrew, working alone, can complete this job in 5 days less than Bill.
How long will it take Bill to make this job?

Solution
Denote number of days for Bill to cover the roof, working himself.
If Andrew works alone, he can complete this job in days.
Thus, in one single day Andrew covers part of the roof area, while Bill covers part of the roof area.
Working together, Andrew and Bill make of the whole work in each single day.
Since they can cover the entire roof in 6 days working together, the equation for the unknown value is as follows:
.

To simplify this equation, multiply both sides by , then transfer all terms from the right side to the left with the opposite signs, then collect common terms and adjust the signs. In this way you get
,
,
,
,
.
You get the quadratic equation. Apply the quadratic formula (see the lesson Introduction into Quadratic Equations) to solve this equation. You get
.
The equation has two roots: and .
The second root does not fit the given conditions (if Bill covers the roof in two days, then Andrew has 2-5=-3 days, what has no sense).

So, the potentially correct solution is : Bill covers the roof in 15 days.
Let us check it. If Bill gets the job done in 15 days, then Andrew makes it in 10 days, working separately.
Since , this solution is correct.

Answer. Bill covers the roof in 15 days.

Problem 3. Filling the liquid reservoir

Two tubes, working together, can fill the reservoir with the liquid in 12 hours.
The larger tube, if works separately, can fill the reservoir in 18 hours less than the smaller tube.
How long will it take to fill the reservoir using the smaller tube only?

Solution
Denote number of hours to fill the reservoir using the smaller tube only.
Then is the time in hours to fill the reservoir using the larger tube only.
Thus, in one single hour the smaller tube will fill part of the reservoir volume, while the larger tube will fill part of the reservoir volume in each single hour.
Working together, these two tubes will fill part of the reservoir volume in each single hour.
Since two tubes working together fill the tank in 12 hours, this gives the equation

.

To simplify this equation, multiply both sides by ,
,
,
,
.
You get the quadratic equation. Apply the quadratic formula (see the lesson Introduction into Quadratic Equations) to solve this equation. You get
.
The equation has two roots: and .
The second root does not fit the given conditions (if smaller tube fills the reservoir in 6 hours, then larger one should make it in 6-18=-12 hours, which has no sense).

So, the potentially correct solution is : smaller tube fills the reservoir in 36 hours.
Let us check it. If smaller tube fills the reservoir in 36 hours, then the larger one makes it in 36-18=18 hours, working separately.
Since , this solution is correct.

Answer. Smaller tube fills the reservoir in 36 hours.