This section covers:
- Quadratic Projectile Problem
- Quadratic Trajectory (Path) Problem
- Optimization of Area Problem
- Maximum Profit and Revenue Problems
- Population Problem
- Linear Increase/Decrease Problem
- Pythagorean Theorem Quadratic Application
- Quadratic Inequality Problem
- Finding Quadratic Equation from Points or a Graph
Quadratic applications are very helpful in solving several types of word problems (other than the bouquet throwing problem), especially where optimization is involved. Again, we can use the vertex to find the maximum or the minimum values, and roots to find solutions to quadratics.
Note that we did a Quadratic Inequality Real World Example here. Note also that we will discuss Optimization Problems using Calculus in the Optimization section here.
Quadratic Projectile Problem:
Jennifer hit a golf ball from the ground and it followed the projectile \(h\left( t \right)=-16{{t}^{2}}+100t\), where \(t\) is the time in seconds, and \(h\) is the height of the ball. Find the highest point that her golf ball reached and also when it hits the ground again. Find a reasonable domain and range for this situation.
Solution:
Note that in this example, we are using the generic equation \(h\left( t \right)=-16{{t}^{2}}+{{v}_{0}}t+{{h}_{0}}\), where, in simplistic terms, the –16 is the gravity (in feet per seconds per seconds), the \({{v}_{0}}\) is the initial velocity (in feet per seconds) and the \({{h}_{0}}\) is the initial height (in feet). (If units are in meters, the gravity is –4.9 meters per second per second).
Since we need to find the highest point of the ball, we need to get the vertex of the parabola. We also need to know when the height \(h\) is back to 0 again. Then we can use these two values to find a reasonable domain and range:
Solve Quadratic Algebraically |
Solve Quadratic with Graphing Calculator |
\(h\left( t \right)=-16{{t}^{2}}+100t\)
To get the vertex, we can use (\(\displaystyle -\frac{b}{{2a}}\), plug \(\displaystyle -\frac{b}{{2a}}\) into the \(t\) to get the \(y\)) to find the coordinates of the vertex, when \(y=a{{t}^{2}}+bt+c\):
\(\displaystyle -\frac{b}{{2a}}=-\frac{{100}}{{-32}}=\,\,\,\,3.125\, \text{seconds}\)
(This is the time, and to get the height, we plug this into \(-16{{t}^{2}}+100t\), and get 156.25 feet.)
To get when the ball hits the ground, we have to set \(-16{{t}^{2}}+100t\) to 0; we get \(t=6.25\) seconds. This is the second root.
This makes sense, since the ball started from the ground, so the parabola is symmetrical around the line of symmetry, which is \(x=3.125\).
The reasonable domain is \(\left[ {0,\,\,6.25} \right]\) and the reasonable range is \(\left[ {0,\,\,156.25} \right]\). |
Note that I used ZOOM 6, ZOOM 0, and ZOOM 3 ENTER a few times so I could see the vertex in the window.
Then I used 2^{nd} TRACE (CALC), 4 (maximum), moved the cursor to the left of the top after “Left Bound?”, hit ENTER, moved the cursor to the right of the top after “Right Bound?”, and then hit ENTER twice to get the vertex.
To get the root, push 2^{nd} TRACE (CALC), and then push 2 for ZERO (or move cursor down to ZERO). The calculator will then say “Left Bound?” Using the cursors, move the cursor anywhere to the left of the zero (where the graph hits the \(x\)-axis) and hit ENTER. We want the zero that is positive. When the calculator says “Right Bound?” move the cursor anywhere to the right of that zero and hit ENTER. The calculator will say “Guess?”. Hit ENTER once more, and you have your zero, which is 6.25 feet. |
(We will discuss projectile motion using parametric equations here in the Parametric Equations section.)
Note that the independent variable represents time, not distance; sometimes parabolas represent the distance on the \(x\)-axis and the height on the \(y\)-axis, and the shapes are similar. Height versus distance would be the path or trajectory of the bouquet, as in the following problem.
Quadratics Trajectory (Path) Problem:
Audrey throws a ball in the air, and the path the ball makes is modeled by the parabola \(y-8=-0.018{{\left( {x-20} \right)}^{2}}\), measured in feet. What is the maximum height the ball reaches, and how far (horizontally) from Audrey does is the ball at its maximum height? How far does the ball travel before it hits the ground?
Solution:
Note that in this problem, the \(x\)-axis is measuring the horizontal distance of the path of the ball, not the time, so when we draw the parabola, it’s a true indication of the trajectory or path of the ball.
Note also that the equation given is in vertex form (if we add 8 to each side), so we can easily get the maximum \(x\) and \(y\) values.
To find out how far the ball travels before it hits the ground, we want to make the \(y\) value 0, so we’ll have \(0=-0.018{{\left( {x-20} \right)}^{2}}+8\). To solve this, we should not expand the square out, but solve using the square root method; this Is much easier.
Let’s solve this problem algebraically:
Quadratic Application Problem |
Solution |
A ball is thrown in the path, measured in feet:
\(y-8=-0.018{{\left( {x-20} \right)}^{2}}\)
What is the maximum height the ball reaches, and how far (horizontally) from Audrey does is the ball at its maximum height? How far does the ball travel before it hits the ground? |
Since the quadratic is already in vertex form (\(y=a{{\left( {x-h} \right)}^{2}}+k\), where \((h,k)\) is the vertex), we can see that the vertex from \(0=-0.018{{\left( {x-20} \right)}^{2}}+8\) is \((20,8)\).
This means that the maximum height (since the parabola opens downward) is 8 feet and it happens 20 feet away from Audrey.
When the ball hits the ground, \(y=0\), so we have \(0=-0.018{{\left( {x-20} \right)}^{2}}+8\). We could expand the binomial and use the quadratic formula, but it’s much easier to use the square root method, since we have a square in the original: \(\begin{align}0&=-0.018{{\left( {x-20} \right)}^{2}}+8\\\frac{{-8}}{{-0.018}}&={{\left( {x-20} \right)}^{2}}\\\pm \sqrt{{\frac{8}{{0.018}}}}&=\sqrt{{{{{\left( {x-20} \right)}}^{2}}}}\\x-20&=\pm 21.08\\x&=41.08\end{align}\) Note that we had to “throw away” the negative part of the solution. The ball will hit the ground 41.08 feet from Audrey. (We could have also used a graphing calculator to solve this problem.) |
Optimization of Area Problem:
Let’s say we are building a cute little rectangular rose garden against the back of our house with a fence around it, but we only have 120 feet of fencing available. What would be the dimensions (length and width) of the garden (with one side attached to the house) to make the area of the garden as large as possible? What is this maximum area? Also, what is a reasonable domain for the width of the garden?
Solution:
Like we did before, let’s set the variables to what the problem is asking. And, as always, let’s draw a picture first when we have any sort of “geometry” word problem.
Let \(w\) equal the width of the garden, and now let’s draw a picture, using the fact that the perimeter is 120. We could have also made the variable the length of the garden, depending on how we draw the picture, but since we have the width on two sides, I thought it might be easier to make the width “\(w\)”:
Picture |
Notes |
Since we know that the perimeter of the rose garden is 120, and it’s rectangular, we know that the 3 sides (2 of which have to be equal), must add up to 120.
If \(w\) equals the width, we know that 2 of the \(w\)’s and the length have to add up to 120. To get the length, let’s use real numbers to see how we should do it. If \(w\) were 10, then we’d have to take away 2 10’s (or 20) from the 120, and the length would be 100.
We can make the length \(120-2w\), and we have the dimensions of the garden in terms of the \(w\), which is what we need. |
Now, we want to maximize the area, and (from Geometry), we know that the area (the \(y\)) is width times length, or \(w(120-2w)\). We want to maximize the area, and when we learned about Quadratics, we learned that the vertex is the maximum \(\boldsymbol{y}\) point, given an \(\boldsymbol{x}\) point, (in our case, a \(\boldsymbol{w}\) point). Area depends on length and width – which makes sense.
Let’s get the vertex for \(A\left( w \right)=w\left( {120-2w} \right)\) both algebraically and using a Graphing Calculator:
Solve Algebraically |
Solve with Graphing Calculator |
\(A\left( w \right)=w\left( {120-2w} \right)\)
To get the vertex, we can use (\(\displaystyle -\frac{b}{{2a}}\), plug \(\displaystyle -\frac{b}{{2a}}\) into the \(w\) to get the \(y\)) to find the coordinates of the vertex, when \(y=a{{w}^{2}}+bw+c\):
\(\displaystyle y=w\left( {120-2w} \right)\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,y=-2{{w}^{2}}+120w\) \(\displaystyle -\frac{b}{{2a}}=-\frac{{120}}{{-4}}=\,\,\,\,30\, \text{feet}\)
(This is the width, and to get the length, we plug this into \(120-2w\), and get 60 feet.)
Now plug in 30 (\(w\)) to get the \(y\), or the area:
\(\left( {30} \right)\left( {120-2\left( {30} \right)} \right)=30\left( {60} \right)=\,\,1800\, \text{fee}{{\text{t}}^{2}}\)
(Note the unit for area is feet squared). |
Note that I used ZOOM 6, ZOOM 0, and ZOOM 3 ENTER a few times so I could see the vertex in the window.
Then I used 2^{nd} TRACE (CALC), 4 (maximum), moved the cursor to the left of the top after “Left Bound?, moved the cursor to the right of the top after “Right Bound?”, and then hit ENTER twice to get the vertex.
Since this is the \(w\) part of the vertex, 30 feet is the width that maximizes the area. The length is \(120-2w\), or 60 feet.
The area is length times width, or the \(y\) part of the vertex, which is \(1800\text{ fee}{{\text{t}}^{2}}\). |
Now, to get the reasonable domain or appropriate domain, we have to think about the values that the width could ever be – to make that garden at all. We know the width has to be positive, which means it has to be greater than zero. But we also know that there is a minus sign in one of the expressions containing the width (the length, which is \(120-2w\)), and this also must be positive. Since \(120-2w>0,\,\,w<60\). A reasonable domain for the width is 0 to 60 feet or, in interval notation, \(\left( {0,60} \right)\).
In this example, if we wanted to find the reasonable range for the area, we would look at the graph, and see that it is \(\left( {0,1800} \right)\). Typically a reasonable range for these types of problems is 0 to the \(\boldsymbol{y}\) portion of the vertex.
Maximum Profit Problem:
The profit from selling local ballet tickets depends on the ticket price. Using past receipts, we find that the profit can be modeled by the function \(p=-15{{x}^{2}}+600x+60\), where \(x\) is the price of each ticket. We want to find the ticket price that gives the maximum profit, and also find that maximum profit.
Solution:
This problem is actually much easier since we are given the formula for the profit, given the price of each ticket.
We simply either graph the function to get the vertex, or use (\(\displaystyle -\frac{b}{{2a}}\), plug \(\displaystyle -\frac{b}{{2a}}\) into the \(x\) to get the \(y\)) to find the coordinates of the vertex:
.\(\displaystyle -\frac{b}{{2a}}=-\frac{{600}}{{2\left( {-15} \right)}}=20,\,\,\,\,\,\,f\left( {20} \right)=-15{{\left( {20} \right)}^{2}}+600\left( {20} \right)+60=6060\)
Since the vertex is \(\left( {20,6060} \right)\), the ticket price should be $20 to maximize profit, and that maximum profit is $6060.
Maximum Revenue Problem:
A popular designer purse sells for $500 and 45,000 are sold a month. The company did some research and realized that for each $20 decrease in price, they can sell 5000 more purses per month. How much should the company charge for the purse so they can maximize monthly revenues?
Solution:
This problem is a little trickier since we can’t really tell from the question what we should set the variables equal to.
But we can think of monthly revenue (what a company makes each month) as “price times number of purses sold”. For example, without making any changes, the monthly revenue is \(\$500\times 45,000=\$22,500,000\) (a lot for purses, but definitely worth it!)
But, with the new information, we know that for each $20 decrease in price, they can sell an increase of 5000 purses per month. For example, if they sell the purses for \(\$500-\$20=\$480\), they would sell \(45,000+5,000=50,000\) purses, for a monthly revenue of $24,000,000, which is more!
Instead of doing all this by hand to find out what we should do to maximize the monthly revenues, we can use algebra to find the maximum monthly revenue by letting x = the number of $20 decreases (and hence sales of 5000 more purses) per month. Then we can find the maximum of our quadratic to get our answers.
Here is our equation:
\(\displaystyle \begin{align}y&=\,\,\,\,\left( {500-20x} \right)\left( {45000+5000x} \right)\\\text{revenue }&=\text{ }\left( {\text{price}} \right)\left( {\text{number sold}} \right)\end{align}\)
We want to find the vertex (maximum) of this quadratic equation, which we can get from a graphing calculator \(\left( {8,28,900,000} \right)\). So 8 is the number of $20 decreases the company can charge a month (which is the number of “batches of 5000 more purses” the company can sell). This means to get the maximum revenues, the company should sell their purses at \(\$500-\$20(8)=\$500-\$160=\$340\), and thus they should be able to sell \(45,000+5000\left( 8 \right)=85,000\) more per month for a maximum profit of $28,900,000.
Bunny Rabbit Population Problem:
The observed bunny rabbit population on an island is given by the function \(p=-.4{{t}^{2}}+130t+1200\), where \(t\) is the time in months since they began observing the rabbits. (a) When is the maximum population attained, (b) what is the maximum population, and (c) when does the bunny rabbit population disappear from the island?
Solution:
This one isn’t too bad, since we are given the equation. The last part of the question is a little different, though, so I’ll go through the graphing calculator steps for this.
(a) and (b) involved finding the vertex again, which will be a maximum. Part (a) involves getting the \(x\) part of the vertex, since the “\(x\)” here is “\(t\)”, which is time. Part (b), the maximum population, will be the \(y\) part of the vertex, since the “\(y\)” here is “\(p\)”, which is population.
From the graphing calculator, we see that the vertex is \(\left( {162.5,11762.5} \right)\). The maximum rabbit population was roughly 11762 rabbits (we can’t have half of a rabbit!) when it was 162.5 months after they began observing the rabbit population. This answers (a) and (b) above.
For (c), we need to see when the graph goes back down to 0; this is when there are no rabbits left on the island. Let’s review how to do this with the graphing calculator:
To answer (c) above, the rabbit population will disappear from the island at around 334 months from when the observations started.
Linear Increase/Decrease Problem:
OK, use your imaginations on this one (sorry!):
Taylor and Miranda are performing on a magic dimension-changing stage that is 20 yards long by 15 yards wide. The length is decreasing linearly (with time) at a rate of 2 yards per hour, and the width is increasing linearly (with time) at a rate of 3 yards per hour. When will the stage have the maximum area, and when will the stage disappear (have an area of 0 square yards)? (Better get off that stage, Taylor and Miranda!)
Solution:
This one’s a little trickier, since we are asking when the stage will be the greatest area, and when it will have an area of 0, yet we are only given distances and rates.
But since we’re finding areas, we need to work with distances only. And we know that \(\text{Distance}=\text{Rate}\times \text{Time}\). Do you see how at time \(t\), the length of the stage is \((20-2t)\) and the width is \((15+3t)\)? Think about it: after one hour, the length of the stage will have decreased by 2 yards, and the width will have increased by 3 yards, so the new stage will be 18 by 18 yards. After two hours, the length will be 16 yards, and the width will be 21 yards, and so on.
Now, let’s find the answers, with and without using the graphing calculator:
Now we need to find when the stage will have no area left. We need to set the equation to 0, or find the rightmost root with the calculator:
Pythagorean Theorem Quadratic Application:
OK, here’s one where you’ll use a bit of Geometry. You probably learned the Pythagorean Theorem awhile back – it’s the one with the right triangle and all the squares in it!
Here is the type of problem you may get:
The hypotenuse of a right triangle is 4 inches longer than one leg and 2 inches longer than the other. Find the dimensions of the triangle. Also, find a reasonable domain for the hypotenuse.
Solution:
This really isn’t an optimization problem, but we’ll see how easy it is to solve with quadratics. Let’s draw a picture first and then use the Pythagorean Theorem:
To get the reasonable domain for the hypotenuse, we know it has to be greater than 0, and since we have minus signs in the expressions for the legs, we have to look at those, too. Both of the legs must have values that are positive, so “\(x-2\)”, and “\(x-4\)” both must be positive. So \(x\) has to be greater than 4 (do you see why?). Therefore, the reasonable domain for the hypotenuse is \(x>4\), or \(\left( {4,\infty } \right)\).
Quadratic Inequality Problem:
You may encounter a problem like this – which is really not too difficult.
You have to remember that the value of the discriminant of quadratics in standard form \(a{{x}^{2}}+bx+c\) has three possibilities:
- \(\displaystyle {{b}^{2}}-4ac=0\) means there is only one real solution.
- \(\displaystyle {{b}^{2}}-4ac>0\) means there are two real solutions.
- \(\displaystyle {{b}^{2}}-4ac<0\) means there are no real solutions, only imaginary solutions.
Here is the problem:
Given \({{x}^{2}}+bx+8\), find the values of \(b\) so there are no real solutions to the quadratic.
Solution:
We see that \(a=2\) and \(c=8\). So we need to find \(b\) such that the discriminant is \(<0\):
\(\displaystyle \begin{array}{c}{{b}^{2}}-4\left( 2 \right)\left( 8 \right)<0\\{{b}^{2}}-64<0\\{{b}^{2}}<64\\\left| b \right|<8\end{array}\)
\(\displaystyle \begin{array}{c}b<8\,\,\,\,\,\,\,\text{and}\,\,\,\,\,b>-8\\b>-8\,\,\,\,\,\,\text{and}\,\,\,\,\,b<8\\-8<x<8\\\left( {-8,8} \right)\end{array}\)
So \(b\) would have to be between –8 and 8 (but can’t include –8 or 8) so there are no real solutions to \(2{{x}^{2}}+bx+8\). Try some numbers for \(b\) to convince yourself that this is correct!
Finding Quadratic Equations from Points or a Graph
We saw how to find a Quadratic Equation from a point and/or graph here in the Solving Quadratics by Factoring and Completing the Square section. Here is one more problem.
Problem:
Emmy throws a dog toy up in the air from 5 feet above the ground. When the toy is 2 feet from the her, the toy reaches a maximum height of 9 feet, and then lands back on the ground 5 feet from her. Find the “\(a\)” (the coefficient of the \({{x}^{2}}\)) for the parabola of the flight of the toy, and write this quadratic equation in vertex form, standard form, and factored form.
Solution:
Quadratic Graph |
Finding Equation |
You can check all three forms by putting them all in a graphing calculator (like in \(\displaystyle {{Y}_{1}},\,\,\,{{Y}_{2}},\,\,\,{{Y}_{3}}\)) to make sure they are all the same parabola!
(You can see from the graph that this looks correct; the roots look like \((0,-1)\) and \((5,0)\).) |
We know the vertex is \((2,9)\), and the \(y\)-intercept is \((0,5)\). “\(a\)” is negative, since the parabola faces downwards. Putting the equation in vertex form would be \(y=a{{\left( {x-2} \right)}^{2}}+9\).
Let’s get the “\(a\)”. Since we only have the vertex form at this point, we can’t use the vertex as the point to plug in (try it and you’ll see why!). Use the \(y\)-intercept \((0,5)\) to plug in to \((x,y)\) to get “\(a\)”: \(\begin{align}y&=a{{\left( {x-2} \right)}^{2}}+9\\5&=a{{\left( {0-2} \right)}^{2}}+9\\5&=4a+9\\-4&=4a;\,\,\,\,\,a=-1\end{align}\)
The vertex form is \(y=-1{{\left( {x-2} \right)}^{2}}+9\) (or \(y=-{{\left( {x-2} \right)}^{2}}+9\)), and the standard form (by multiplying it out) is \(y=-{{x}^{2}}+4x+5\).
We can factor this to get the factored form, which is \(y=-\left( {x-5} \right)\left( {x+1} \right)\) (take the negative out first). |
Learn these rules, practice, practice, practice, and you’ll rock at math!
On to Solving Absolute Value Equations and Inequalities – you’re ready!