The profit P, in dollars, gained by selling X computers is modeled by the equation P+-5X^2+1000x+5000. How many computers must be sold to obtain a profit of 55,000
well, you have the formula, so just solve
-5x^2 + 1000x + 5000 = 55000
x^2 - 200x + 1000 = 0
(x-100)^2 = 0
x = 100
To find the number of computers that must be sold to obtain a profit of $55,000, we need to solve the equation P = -5X^2 + 1000X + 5000 for X.
Given that P = 55,000, we can substitute it in the equation:
55,000 = -5X^2 + 1000X + 5000
Rearranging the equation:
-5X^2 + 1000X + 5000 - 55,000 = 0
-5X^2 + 1000X - 50,000 = 0
Now, we can solve the quadratic equation. There are a few ways to solve it, but for this example, let's use the quadratic formula:
X = (-b ± √(b^2 - 4ac)) / (2a)
Using the values from our equation:
a = -5
b = 1000
c = -50,000
Plugging in these values, we get:
X = ( -1000 ± √(1000^2 - 4(-5)(-50,000))) / (2(-5))
Calculating further:
X = ( -1000 ± √(1,000,000 - 1,000,000)) / (-10)
X = ( -1000 ± √0) / (-10)
Since √0 = 0, we have:
X = ( -1000 ± 0) / (-10)
Simplifying further:
X = -1000 / -10
X = 100
Therefore, to obtain a profit of $55,000, 100 computers must be sold.
To find the number of computers that must be sold to obtain a profit of $55,000, we need to solve the equation P = 55,000.
The given equation, P = -5X^2 + 1000X + 5000, represents the profit gained by selling X computers.
Substitute the value of 55,000 for P in the equation:
55,000 = -5X^2 + 1000X + 5000
Now, we have a quadratic equation in the form of ax^2 + bx + c = 0. Rearrange the equation to set it equal to zero:
-5X^2 + 1000X + 5000 - 55,000 = 0
-5X^2 + 1000X - 50,000 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Factoring might not always be possible or practical, so let's use the quadratic formula:
X = (-b ± √(b^2 - 4ac)) / (2a)
For our equation -5X^2 + 1000X - 50,000 = 0, the values of a, b, and c are:
a = -5, b = 1000, and c = -50,000.
Plug these values into the quadratic formula:
X = (-(1000) ± √((1000)^2 - 4(-5)(-50,000))) / (2(-5))
Simplify the equation further:
X = (-1000 ± √(1,000,000 + 1,000,000)) / (-10)
X = (-1000 ± √2,000,000) / (-10)
X = (-1000 ± 1414.21) / (-10)
Now, we have two solutions:
X1 = (-1000 + 1414.21) / (-10)
X1 = 414.21 / (-10)
X1 = -41.42
X2 = (-1000 - 1414.21) / (-10)
X2 = -2414.21 / (-10)
X2 = 241.42
Since the number of computers cannot be negative, we discard the negative solution. Therefore, the number of computers that must be sold to obtain a profit of $55,000 is approximately 241.42, rounded to the nearest whole number.