Sure, let's break down each part of the question and provide detailed solutions for each step.
### i. Express the total manufacturing cost as a function of [tex]\( t \)[/tex]
Given:
- The cost function [tex]\( C(q) = q^2 + q + 900 \)[/tex]
- The quantity function [tex]\( q(t) = 25t \)[/tex]
To express the total manufacturing cost [tex]\( C \)[/tex] as a function of [tex]\( t \)[/tex]:
1. Substitute [tex]\( q(t) \)[/tex] into the cost function [tex]\( C \)[/tex].
[tex]\[
C(q(t)) = C(25t)
\][/tex]
2. Replace [tex]\( q \)[/tex] with [tex]\( 25t \)[/tex] in [tex]\( C(q) \)[/tex]:
[tex]\[
C(25t) = (25t)^2 + 25t + 900
\][/tex]
3. Simplify the expression:
[tex]\[
C(25t) = 625t^2 + 25t + 900
\][/tex]
Therefore, the total manufacturing cost as a function of [tex]\( t \)[/tex] is:
[tex]\[
C(t) = 625t^2 + 25t + 900
\][/tex]
### ii. How much will have been spent on production by the end of the 3rd hour
To find the cost at the end of the 3rd hour, substitute [tex]\( t = 3 \)[/tex] into the cost function [tex]\( C(t) \)[/tex]:
[tex]\[
C(3) = 625(3)^2 + 25(3) + 900
\][/tex]
By evaluating the above expression, we get:
[tex]\[
C(3) = 6600
\][/tex]
Therefore, the cost at the end of the 3rd hour is 6600 Kshs.
### iii. When will the cost be minimum
To find when the cost will be minimum, we need to take the derivative of the cost function [tex]\( C(q) \)[/tex] with respect to [tex]\( q \)[/tex] and set it to zero:
1. The cost function is [tex]\( C(q) = q^2 + q + 900 \)[/tex].
2. Take the first derivative [tex]\( C'(q) \)[/tex]:
[tex]\[
C'(q) = 2q + 1
\][/tex]
3. Set the derivative equal to zero to find the critical points:
[tex]\[
2q + 1 = 0
\][/tex]
4. Solve for [tex]\( q \)[/tex]:
[tex]\[
q = -\frac{1}{2}
\][/tex]
Therefore, the cost is minimum when [tex]\( q = -\frac{1}{2} \)[/tex].
### iv. What is the minimum value
To find the minimum value of the cost function [tex]\( C(q) \)[/tex], substitute [tex]\( q = -\frac{1}{2} \)[/tex] back into the cost function [tex]\( C(q) \)[/tex]:
[tex]\[
C\left(-\frac{1}{2}\right) = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + 900
\][/tex]
Evaluating the above expression yields:
[tex]\[
C\left(-\frac{1}{2}\right) = \frac{3599}{4}
\][/tex]
Therefore, the minimum value of the cost function is [tex]\( \frac{3599}{4} \)[/tex] Kshs.
