An artist makes a profit of (500p - p^2) dollars from selling p paintings. What is the fewest number of paintings the artist can sell to make a profit of at least $60,000?
Don't know how to even set it up
just use what you are given:
500p - p^2 >= 60000
Now, just from what you know about parabolas, you know that the parabola opens downward. So, the portion of the curve above a horizontal line (here, y=60000) will lie between the roots of
500p - p^2 - 60000 = 0
The roots are 200, 300.
So, selling anywhere between 200 and 300 paintings will result in a profit of at least 60,000.
See
http://www.wolframalpha.com/input/?i=500p+-+p^2+%3E%3D+60000
150
To find the fewest number of paintings the artist must sell to make a profit of at least $60,000, we can set up an inequality.
The profit equation is given as 500p - p^2, where p represents the number of paintings sold.
We want to find the minimum number of paintings that will result in a profit of at least $60,000. So we set up the following equation:
500p - p^2 ≥ 60,000
To solve this inequality, we can rearrange it by moving all terms to one side:
p^2 - 500p + 60,000 ≤ 0
Now, we can factorize the quadratic equation:
(p - 200)(p - 300) ≤ 0
To find the values of p that satisfy this inequality, we should consider the signs of the expressions (p - 200) and (p - 300):
When (p - 200) ≤ 0 and (p - 300) ≥ 0, p ≤ 200.
When (p - 200) ≥ 0 and (p - 300) ≤ 0, 200 ≤ p ≤ 300.
When (p - 200) ≤ 0 and (p - 300) ≤ 0, p ≥ 300.
To determine the fewest number of paintings, we take the smallest possible value of p, which is 200.
Therefore, the artist must sell at least 200 paintings to make a profit of at least $60,000.
To find the fewest number of paintings the artist can sell to make a profit of at least $60,000, we need to set up an inequality using the given profit equation.
The profit equation is:
Profit = 500p - p^2
Since the artist wants to make a profit of at least $60,000, we can write the inequality as:
500p - p^2 ≥ 60,000
To solve this inequality, we can rearrange it to its standard form:
p^2 - 500p + 60,000 ≤ 0
Now, let's solve this quadratic inequality by factoring:
(p - 200)(p - 300) ≤ 0
This inequality states that the product of two factors, (p - 200) and (p - 300), is less than or equal to zero.
To find the solution, we need to consider the signs of the factors:
For (p - 200) to be negative, p must be less than 200.
For (p - 300) to be negative, p must be less than 300.
Since we are looking for the solution where the inequality is less than or equal to zero, we'll consider two scenarios:
1. Both factors are negative:
p < 200 and p < 300
2. One factor is zero:
p = 200 or p = 300
Now, we can determine the potential solutions for p:
1. When p < 200:
The artist should sell fewer than 200 paintings.
2. When p < 300:
The artist should sell fewer than 300 paintings.
3. When p = 200:
The artist should sell exactly 200 paintings.
4. When p = 300:
The artist should sell exactly 300 paintings.
Therefore, the fewest number of paintings the artist can sell to make a profit of at least $60,000 is 200 paintings.