Math Problem Statement
Solution
Let's solve the questions provided in the image step by step.
Given:
- The total cost of manufacturing units is:
- The number of units manufactured after hours is modeled by:
a. Express the total manufacturing cost as a function of .
To express the cost as a function of , we substitute into the cost function :
Simplifying:
So, the total manufacturing cost as a function of is:
b. How much will have been spent on production by the end of the third hour?
We substitute into the cost function :
Calculating step by step:
So, the total cost spent by the end of the third hour is $6,600.
c. When will the total manufacturing cost reach $11,000?
We set and solve for :
First, subtract 11000 from both sides:
Now, divide through by 25 to simplify:
We can solve this quadratic equation using the quadratic formula: where , , and .
Let's calculate this value.The solutions to the quadratic equation are:
Since negative time doesn't make sense in this context, we conclude that the total manufacturing cost will reach $11,000 after 4 hours.
Would you like any further details or clarification?
Here are 5 related questions for further exploration:
- How would changes in the production rate affect the total cost function?
- What would the cost be after 5 hours of production?
- Can you derive a general formula for the number of hours required to reach a given total cost?
- How much of the total cost is fixed, and how much depends on the units produced?
- How would the cost equation change if the fixed cost of 900 dollars increased by 10%?
Tip: When dealing with quadratic equations, always check if dividing the equation by a common factor simplifies calculations.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Functions
Formulas
Total cost function: C(q) = q^2 + q + 900
Units produced function: q(t) = 25t
Quadratic equation formula: ax^2 + bx + c = 0
Theorems
Quadratic formula
Suitable Grade Level
Grades 10-12
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