after doing some market research, crunchy cookie inc. has found that their profit on cookies after spending…
after doing some market research, crunchy cookie inc. has found that their profit on cookies after spending $x$ amount on advertising can be modeled by $p(x)=5000 + 1000x-5x^{2}$. both $x$ and $p(x)$ are measured in thousands of dollars.\na. what is the most profit that crunchy cookie inc. can make?\nb. how much would they have to spend on advertising to achieve this result?\nc. is there ever a time when crunchy cookie inc. will make no profit? explain why or why not.\n4. the graph of the function $y = f(x)$ models the relationship between two variables $x$ and $y$.\na. describe what is happening to the values of $y$ as $x$ increases. be as precise as possible.\nb. what context could the independent and dependent variables of this graph depict? explain your choice.
Answer
Explanation:
Step1: Identify the profit - function type
The profit function $P(x)=5000 + 1000x-5x^{2}$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a=-5$, $b = 1000$, and $c = 5000$.
Step2: Find the x - value of the vertex
The x - value of the vertex of a quadratic function $y=ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. Substituting $a=-5$ and $b = 1000$ into the formula, we get $x=-\frac{1000}{2\times(-5)}=\frac{-1000}{-10}=100$.
Step3: Find the maximum profit
Substitute $x = 100$ into the profit function $P(x)=5000 + 1000x-5x^{2}$. $P(100)=5000+1000\times100 - 5\times100^{2}$ $=5000 + 100000-5\times10000$ $=5000 + 100000 - 50000$ $=55000$.
Step4: Check for zero - profit
Set $P(x)=0$, so $5000 + 1000x-5x^{2}=0$. Divide through by 5 to simplify: $1000 + 200x-x^{2}=0$, or $x^{2}-200x - 1000=0$. The discriminant of a quadratic equation $ax^{2}+bx + c = 0$ is $\Delta=b^{2}-4ac$. Here, $a = 1$, $b=-200$, and $c=-1000$. $\Delta=(-200)^{2}-4\times1\times(-1000)=40000 + 4000=44000>0$. So, there are two real - valued solutions for $x$, which means there are times when the company will make no profit.
For the second function $y = f(x)$:
Step5: Analyze the function behavior
As $x$ increases, the function $y = f(x)$ first increases at an increasing rate (concave - up) and then increases at a decreasing rate (concave - down).
Step6: Suggest a context
The independent variable $x$ could represent time, and the dependent variable $y$ could represent the growth of a population. In the beginning, the population grows exponentially (concave - up) due to abundant resources, and then as resources become limited, the growth rate slows down (concave - down).
Answer:
a. The most profit is $55000$ thousand dollars. b. They have to spend $100$ thousand dollars on advertising to achieve this result. c. Yes, because the discriminant of the quadratic equation $x^{2}-200x - 1000=0$ is positive, so there are two real - valued solutions for the amount of advertising spend when the profit is zero. 4a. As $x$ increases, the function $y = f(x)$ first increases at an increasing rate (concave - up) and then increases at a decreasing rate (concave - down). 4b. The independent variable $x$ could be time and the dependent variable $y$ could be the size of a population. Initially, with abundant resources, the population grows exponentially (concave - up), and as resources become scarce, the growth rate slows (concave - down).