7.7: Solve applications by factoring
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There are two types of applications we will discuss in this section. There are many more types of problems we can model by a trinomial equation, but we will only discuss integer and rectangle problems for now. There’s more to come in a later chapter.
Integer Problems
The product of two positive integer numbers is \(48\) and the sum of the same two numbers is \(14\). Find the numbers.
Solution
First, we recall the method of substitution in system of equation in two variables. Recall, we solved for one variable in one equation, then substituted the expression into the second equation. We apply this method for integer problems. Let’s set up the system. Let \(x\) and \(y\) be the two positive integers:
\[\begin{aligned}xy&=48 \\ x+y&=14\end{aligned}\]
Taking the second equation and rewriting it as \(y = 14−x\), we substitute \(y\) into the first equation:
\[\begin{aligned}x\color{blue}{y}\color{black}{}&=48 \\ x\color{blue}{(14-x)}\color{black}{}&=48\end{aligned}\]
Now, we can solve.
\[\begin{array}{rl}x\color{blue}{(14-x)}\color{black}{}=48&\text{Distribute} \\ 14x-x^2=48&\text{Rewrite with zero on the right side} \\ -x^2+14x-48=0&\text{Multiply each term by }-1 \\ x^2-14x+48=0&\text{Factor} \\ (x-6)(x-8)=0&\text{Apply the zero product rule} \\ x-6=0\quad\text{or}\quad x-8=0&\text{Solve} \\ x=6\quad\text{or}\quad x=8&\text{Solution}\end{array}\nonumber\]
Since both solutions are positive, then the numbers are \(6\) and \(8\).
William Horner, a British mathematician from the late \(18^{\text{th}}\) century/early \(19^{\text{th}}\) century, is credited with a method for solving simultaneous equations. However, Chinese mathematician Chu Shih-chieh in 1303 solved these equations with exponents as high as \(14\).
Rectangles Using the Area
The length of a rectangle is \(3\) more inches than the width. If the area is \(40\) square inches, what are the dimensions?
Solution
First, we need to recall the formula for the area of a rectangle:
\[A=\ell\cdot w\nonumber\]
We use this formula to model a trinomial equation. We know that the length of the rectangle is \(3\) more inches than the width:
\[\ell =3+w\nonumber\]
Next, we are given that the area of this rectangle is \(40\) square inches: \(A = 40\). Let’s model this information into a trinomial equation:
\[\begin{array}{rl}A=\ell\cdot w&\text{Replace }\ell\text{ with }3+w\text{ and }A=40 \\ 40=(3+w)\cdot w&\text{Distribute} \\ 40=3w+w^2&\text{Rewrite with zero on the right} \\ -w^2-3w+40=0&\text{Multiply each term by }-1\\ w^2+3w-40=0\end{array}\nonumber\]
Next, we solve the equation by factoring:
\[\begin{array}{rl}w^2+3w-40=0&\text{Factor} \\ (w-5)(w+8)=0&\text{Apply the zero product rule} \\ w-5=0\quad\text{or}\quad w+8=0&\text{Solve} \\ w=5\quad\text{or}\quad w=-8&\text{Solution}\end{array}\nonumber\]
Since we have a rectangle and are finding the length and width of the rectangle, then we omit any negative solutions because length and width cannot be negative. Hence, we omit \(w = −8\), and obtain a width of \(5\) inches and a length of \(8\) inches \((ℓ = 3 + 5)\).
Factoring Applications with Functions
A rocket is launched at \(t = 0\) seconds. Its height, in feet, above sea-level, as a function of time, \(t\), is given by \(h(t) = −16t^2 + 144t + 352\). When does the rocket hit the ground after it is launched?
Solution
A rocket reaches the ground after it is launched when there is no distance between the rocket and the ground. Hence, the height between the rocket and ground is \(0\) feet. We need to find \(t\) when \(h(t)=0\).
\[\begin{array}{rl}h(t)=-16t^2+144t+352&\text{Replace }h(t)\text{ with zero} \\ 0=-16t^2+144t+352&\text{Factor the GCF }-16 \\ 0=-16(t^2-9t-22)&\text{Divide each side by }-16 \\ 0=t^2-9t-22&\text{Factor} \\ 0=(t-11)(t+2)&\text{Apply the zero product rule} \\ t-11=0\quad\text{or}\quad t+2=0&\text{Solve} \\ t=11\quad\text{or}\quad t=-2&\text{Solutions}\end{array}\nonumber\]
With applications, we omit answers that are not reasonable and since we are trying to obtain the time it takes for the rocket to hit the ground, we should omit the solution \(t = −2\). Thus, it will take \(11\) seconds for the rocket to hit the ground.
The profit for a certain commodity, \(n\), where \(n\) is in units, is given by the function \[P(n)=-25n^2+400n+1425\nonumber\]
At the break-even point, the profit is zero, i.e., \(P(n) = 0\). Find the number of units where the break-even point is located, i.e., find \(n\) when \(P(n) = 0\).
Solution
The break-even point is when the profit is zero, i.e., when \(P(n) = 0\). We need to set \(P(n) = 0\) and solve for \(n\).
\[\begin{array}{rl}P(n)=-25n^2+400n+1425&\text{Replace }P(n)\text{ with zero} \\ 0=-25n^2+400n+1425&\text{Factor the GCF }-25 \\ 0=-25(n^2-16n-57)&\text{Divide each side by }-25 \\ 0=n^2-16n-57&\text{Factor} \\ 0=(n-19)(n+3)&\text{Apply the zero product rule} \\ n-19=0\quad\text{or}\quad n+3=0&\text{Solve} \\ n=19\quad\text{or}\quad n=-3&\text{Solutions}\end{array}\nonumber\]
With applications, we omit answers that are not reasonable and since we are trying to obtain the number of units where the break-even point is located, then we should omit the solution \(n = −3\). Thus, the break-even point is located after \(19\) units are sold and produce
Solve Applications by Factoring Homework
The product of two positive integer numbers is \(104\) and the sum of the same two numbers is \(21\). Find the numbers.
The product of two positive integer numbers is \(50\) and the sum of the same two numbers is \(15\). Find the numbers.
The product of two positive integer numbers is \(48\) and the sum of the same two numbers is \(14\). Find the numbers.
The product of two positive integer numbers is \(30\) and the sum of the same two numbers is \(11\). Find the numbers.
The product of two positive integer numbers is \(44\) and the sum of the same two numbers is \(15\). Find the numbers.
The product of two positive integer numbers is \(18\) and the sum of the same two numbers is \(11\). Find the numbers.
The product of two positive integer numbers is \(28\) and the sum of the same two numbers is \(11\). Find the numbers.
The product of two positive integer numbers is \(18\) and the sum of the same two numbers is \(9\). Find the numbers.
The product of two positive integer numbers is \(54\) and the sum of the same two numbers is \(15\). Find the numbers.
The length of a rectangle is \(6\) inches longer than it is wide. If the area is \(160\) square inches, what are the dimensions of the rectangle?
The length of a rectangle is \(4\) inches longer than it is wide. If the area is \(117\) square inches, what are the dimensions of the rectangle?
The length of a rectangle is \(6\) inches longer than it is wide. If the area is \(91\) square inches, what are the dimensions of the rectangle?
The length of a rectangle is \(4\) inches longer than it is wide. If the area is \(45\) square inches, what are the dimensions of the rectangle?
The length of a rectangle is \(7\) inches longer than it is wide. If the area is \(98\) square inches, what are the dimensions of the rectangle?
The length of a rectangle is \(6\) inches longer than it is wide. If the area is \(40\) square inches, what are the dimensions of the rectangle?
The length of a rectangle is \(7\) inches longer than it is wide. If the area is \(144\) square inches, what are the dimensions of the rectangle?
The length of a rectangle is \(4\) inches longer than it is wide. If the area is \(96\) square inches, what are the dimensions of the rectangle?
The length of a rectangle is \(7\) inches longer than it is wide. If the area is \(60\) square inches, what are the dimensions of the rectangle?
The length of a rectangle is \(3\) inches longer than it is wide. If the area is \(88\) square inches, what are the dimensions of the rectangle?
A rocket is launched at \(t = 0\) seconds. Its height, in feet, above sea-level, as a function of time, \(t\), is given by \[h(t) = −16t^2 + 64t + 192\nonumber\] When does the rocket hit the ground after it is launched?
A rocket is launched at \(t = 0\) seconds. Its height, in feet, above sea-level, as a function of time, \(t\), is given by \[h(t) = −16t^2 + 48t + 448\nonumber\] When does the rocket hit the ground after it is launched?
The profit for a certain commodity, \(n\), where \(n\) is in units, is given by the function \[P(n) = −25n^2 + 325n + 750\nonumber\] At the break-even point, the profit is zero, i.e., \(P(n) = 0\). Find the number of units where the break-even point is located, i.e., find \(n\) when \(P(n) = 0\).
The profit for a certain commodity, \(n\), where \(n\) is in units, is given by the function \[P(n) = −25n^2 + 325n + 1200\nonumber\] At the break-even point, the profit is zero, i.e., \(P(n) = 0\). Find the number of units where the break-even point is located, i.e., find \(n\) when \(P(n) = 0\).