A collection of math questions are present in the image. These questions cover topics such as: Direction cosines, Deriving functions, Profit maximization, Properties of equations a... A collection of math questions are present in the image. These questions cover topics such as: Direction cosines, Deriving functions, Profit maximization, Properties of equations and Geometry.
Understand the Problem
The image contains several math questions. Specifically, it includes problems related to direction cosines, function derivatives, profit maximization, geometry (circumcenter of a triangle) and proving properties about lines and equations.
Answer
The company should manufacture $x = 75$ items. The maximum profit is $P(75) = 4625$.
Answer for screen readers
The company should manufacture 75 items to get the maximum profit. The maximum profit is 4625.
Steps to Solve
- Write the profit function
The profit function is given as: $P(x) = (150 - x)x - 1000$
- Expand the profit function
Expanding simplifies the derivative: $P(x) = 150x - x^2 - 1000$
- Find the first derivative of the profit function
To find the maximum profit, we need to find the critical points by taking the derivative and setting it to zero: $P'(x) = \frac{d}{dx}(150x - x^2 - 1000) = 150 - 2x$
- Set the first derivative equal to zero and solve for $x$
$150 - 2x = 0$ $2x = 150$ $x = 75$
This value of $x$ is a critical point. To confirm this is a maximum, we can use the second derivative test.
- Find the second derivative of the profit function
$P''(x) = \frac{d^2}{dx^2}(150x - x^2 - 1000) = \frac{d}{dx}(150 - 2x) = -2$
- Check the sign of the second derivative
Since $P''(x) = -2 < 0$, the profit function is concave down, and thus $x = 75$ corresponds to a maximum.
- Calculate the maximum profit
Substitute $x = 75$ into the profit function: $P(75) = (150 - 75)(75) - 1000 = (75)(75) - 1000 = 5625 - 1000 = 4625$
The company should manufacture 75 items to get the maximum profit. The maximum profit is 4625.
More Information
The profit function is a quadratic with a negative leading coefficient, indicating a maximum. The second derivative confirms this.
Tips
A common mistake is not expanding the profit function before taking the derivative, which can lead to errors. Also, forgetting to check the second derivative to confirm that the critical point corresponds to a maximum (rather than a minimum or inflection point).
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