Math Problem Statement

At a certain factory, the total cost of manufacturing q units during the daily production run is C(q) = q^2 + q + 900 dollars. On a typical workday, the number of units manufactured during the first t hours of a production run can be modeled by the function q(t) = 25t. a. Express the total manufacturing cost as a function of t. b. How much will have been spent on production by the end of the third hour? c. When will the total manufacturing cost reach $11,000?

Solution

Let's solve the questions provided in the image step by step.

Given:

  1. The total cost of manufacturing qq units is: C(q)=q2+q+900C(q) = q^2 + q + 900
  2. The number of units manufactured after tt hours is modeled by: q(t)=25tq(t) = 25t

a. Express the total manufacturing cost as a function of tt.

To express the cost CC as a function of tt, we substitute q(t)=25tq(t) = 25t into the cost function C(q)C(q):

C(t)=C(q(t))=(25t)2+(25t)+900C(t) = C(q(t)) = (25t)^2 + (25t) + 900

Simplifying:

C(t)=625t2+25t+900C(t) = 625t^2 + 25t + 900

So, the total manufacturing cost as a function of tt is: C(t)=625t2+25t+900C(t) = 625t^2 + 25t + 900

b. How much will have been spent on production by the end of the third hour?

We substitute t=3t = 3 into the cost function C(t)C(t):

C(3)=625(3)2+25(3)+900C(3) = 625(3)^2 + 25(3) + 900

Calculating step by step:

C(3)=625×9+75+900C(3) = 625 \times 9 + 75 + 900 C(3)=5625+75+900C(3) = 5625 + 75 + 900 C(3)=6600C(3) = 6600

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 C(t)=11000C(t) = 11000 and solve for tt:

625t2+25t+900=11000625t^2 + 25t + 900 = 11000

First, subtract 11000 from both sides:

625t2+25t+90011000=0625t^2 + 25t + 900 - 11000 = 0 625t2+25t10100=0625t^2 + 25t - 10100 = 0

Now, divide through by 25 to simplify:

25t2+t404=025t^2 + t - 404 = 0

We can solve this quadratic equation using the quadratic formula: t=b±b24ac2at = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} where a=25a = 25, b=1b = 1, and c=404c = -404.

Let's calculate this value.The solutions to the quadratic equation are:

t=4.0ort=4.04t = 4.0 \quad \text{or} \quad t = -4.04

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:

  1. How would changes in the production rate q(t)q(t) affect the total cost function?
  2. What would the cost be after 5 hours of production?
  3. Can you derive a general formula for the number of hours required to reach a given total cost?
  4. How much of the total cost is fixed, and how much depends on the units produced?
  5. 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