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At certain factory, the marginal cost is RM 3(𝑞 − 4)ଶ per unit when the level of...
Apr 30, 2024
At certain factory, the marginal cost is RM 3(𝑞 − 4)ଶ per unit when the level of production is 𝑞 units. By how much will the total manufacturing cost increase if the level of production is raised from 6 units to 10 units?
Generated Graph
Solution by Steps
step 1
To find the increase in total manufacturing cost, we need to integrate the marginal cost function from the initial production level to the final production level
step 2
The marginal cost function is given by 3(q−4)23(q - 4)^2. We integrate this function from q=6q = 6 to q=10q = 10
step 3
The definite integral of the marginal cost function from q=6q = 6 to q=10q = 10 is calculated as ∫6103(q−4)2 dq\int_{6}^{10} 3(q - 4)^2 \, dq
step 4
According to the asksia-ll calculation list, the result of the integral is 208208
Answer
The total manufacturing cost will increase by RM 208 when the level of production is raised from 6 units to 10 units.
Key Concept
Definite Integral of Marginal Cost Function
Explanation
The definite integral of the marginal cost function over the interval from the initial to the final production levels gives the increase in total cost.
A manufacturer determines that 𝑡 months after introducing a new product, the company’s sales will be RM 𝑆(𝑡) thousand, where 𝑆(𝑡) = 750𝑡 √4𝑡 ଶ + 25 What are the average monthly sales of the company over the first 6 months after the introduction of the new product?
Solution by Steps
step 1
To find the average monthly sales over the first 6 months, we need to calculate the definite integral of the sales function S(t)S(t) from t=0t=0 to t=6t=6
step 2
The definite integral of S(t)S(t) from t=0t=0 to t=6t=6 is given by the asksia-ll calculator as 15001500
step 3
To find the average value of the function over the interval from t=0t=0 to t=6t=6, we divide the definite integral by the length of the interval, which is 6−0=66-0=6
step 4
The average monthly sales are therefore 15006=250\frac{1500}{6} = 250
Answer
The average monthly sales of the company over the first 6 months after the introduction of the new product are RM 250 thousand.
Key Concept
Average Value of a Function over an Interval
Explanation
The average value of a continuous function over a closed interval [a, b] is given by 1b−a∫abf(x)dx\frac{1}{b-a} \int_{a}^{b} f(x) dx. In this case, the function represents the company's sales, and the interval is the first 6 months after the product launch.
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