Practise the method.
Bounded regions, production costs and recovering model parameters.
Example 1
A process adds material at rate r(t)=24e−t grams per hour, for t≥0. Initially Q(0)=40 grams are present. No material is removed. Use exact expressions until final rounding.
Find the mass added in the first 4 hours. Give your answer to 3 decimal places.
Show worked answer
Answer: 23.560
- The mass added is ∫04(24e−t)dt.
- Integrating gives [−(124)e−t] from 0 to 4.
- The added mass is (124)(1−e−4)=23.5604246667 grams. The initial mass is not part of the added mass.
- Therefore the answer is 23.560 to 3 decimal places.
Example 2
A production model has marginal cost C′(x)=6x+b dollars per unit, for 0≤x≤3. The fixed cost is C(0)=40 dollars and C(1)=85 dollars. The constant b is unknown.
a Find b.
Find the additional cost of increasing production from x=2 to x=3.
Show worked answer
Answer: 57
- Use C(1) to obtain b=145−3=42.
- The additional cost is ∫23(6x+42)dx.
- Evaluate [3x2+42x] between 2 and 3:153−96=57 dollars.
- The fixed cost cancels because it is paid once, not again for the extra batch.
Example 3
A production model has marginal cost C′(x)=−4x+b dollars per unit, for 0≤x≤12. The fixed cost is C(0)=50 dollars and C(4)=402 dollars. The constant b is unknown.
a Find b.
b Find the additional cost of increasing production from x=8 to x=12.
Find x in 0≤x≤12 when C(x)=554 dollars. The model allows non-integer production quantities.
Show worked answer
Answer: 6
- The recovered model is C(x)=−2x2+96x+50.
- Set it equal to 554:−2x2+96x−504=0.
- The discriminant is 962−4(−2)(−504)=5184.
- The quadratic formula gives x=−4−96±√5184, so x=6 or x=42.
- Only x=6 is in [0,12]. Also C′(x) stays positive on this interval, so the valid solution is unique.