Annual profit in thousands of dollars is given by the function, P(x) = -.1x2 + 50x - 300, where x is the number of items sold, x ≥ 0.
describe the meaning of the number -.1 in the formula, in terms of its meaning in relation to the profit.
describe the meaning of the number -300 in the formula, in terms of its meaning in relation to the profit.
find the profit for 5 different values of x
To explain the meaning of the numbers in the given profit function P(x) = -.1x^2 + 50x - 300, we need to understand the impact of each term on the profit.
1. The coefficient -.1:
The coefficient -.1 in the function represents the quadratic term -0.1x^2. This term indicates that the profit function is a quadratic function that traces a concave-downward parabola. Since the coefficient is negative, it means that the profit generated decreases as the number of items sold (x) increases. In other words, as more items are sold, the rate of profit growth decreases, reflecting diminishing returns on sales.
2. The constant -300:
The constant term -300 represents a fixed cost or an initial investment that needs to be covered before any profit is achieved. This constant value of -300 implies that even if no items are sold (x=0), there is still an incurred cost of $300, which reduces the overall profit. Essentially, the constant term accounts for the initial expenses or non-revenue-related costs.
Now, let's find the profit for five different values of x using the given profit function P(x) = -.1x^2 + 50x - 300:
For x = 1:
P(1) = -0.1(1)^2 + 50(1) - 300
P(1) = -0.1 + 50 - 300
P(1) = 49.9 - 300
P(1) = -250.1
For x = 2:
P(2) = -0.1(2)^2 + 50(2) - 300
P(2) = -0.4 + 100 - 300
P(2) = 99.6 - 300
P(2) = -200.4
For x = 3:
P(3) = -0.1(3)^2 + 50(3) - 300
P(3) = -0.9 + 150 - 300
P(3) = 149.1 - 300
P(3) = -150.9
For x = 4:
P(4) = -0.1(4)^2 + 50(4) - 300
P(4) = -1.6 + 200 - 300
P(4) = 198.4 - 300
P(4) = -101.6
For x = 5:
P(5) = -0.1(5)^2 + 50(5) - 300
P(5) = -2.5 + 250 - 300
P(5) = 247.5 - 300
P(5) = -52.5
Therefore, the profits for the five different values of x are as follows:
For x=1: -$250.1
For x=2: -$200.4
For x=3: -$150.9
For x=4: -$101.6
For x=5: -$52.5