q=30-0.3p &cost function c=2qpowe2+20q+10
To find the value of p that minimizes the cost function c, we need to find the derivative of the cost function with respect to p and set it equal to 0. Let's go step by step.
First, let's rewrite the equation q = 30 - 0.3p in terms of p:
q = 30 - 0.3p
0.3p = 30 - q
p = (30 - q) / 0.3
Now, let's substitute this expression for q in the cost function c:
c = 2q^2 + 20q + 10
c = 2((30 - q) / 0.3)^2 + 20((30 - q) / 0.3) + 10
Next, we differentiate the cost function c with respect to p:
dc/dp = d/dp [2((30 - q) / 0.3)^2 + 20((30 - q) / 0.3) + 10]
To make it simpler, we can multiply the entire expression by (0.3)^2:
dc/dp = (0.3)^2 * d/dp [2((30 - q) / 0.3)^2 + 20((30 - q) / 0.3) + 10]
Now, we can use the chain rule to differentiate each term separately:
dc/dp = (0.3)^2 * [d/dp (2((30 - q) / 0.3)^2) + d/dp (20((30 - q) / 0.3)) + d/dp (10)]
Differentiating each term:
d/dp (2((30 - q) / 0.3)^2) = 4((30 - q) / 0.3) * d/dp ((30 - q) / 0.3)
d/dp (20((30 - q) / 0.3)) = 20 * d/dp ((30 - q) / 0.3)
d/dp (10) = 0 (since it is a constant)
Now, substituting back q = (30 - p) / 0.3:
dc/dp = (0.3)^2 * [4((30 - q) / 0.3) * d/dp ((30 - q) / 0.3) + 20 * d/dp ((30 - q) / 0.3) + 0]
dc/dp = 0.09 * [4((30 - ((30 - p) / 0.3)) / 0.3) * (d/dp ((30 - ((30 - p) / 0.3)) / 0.3)) + 20 * (d/dp ((30 - ((30 - p) / 0.3)) / 0.3))]
Now, we can simplify this expression to find the derivative of the cost function with respect to p:
dc/dp = 0.09 * [4((30 - ((30 - p) / 0.3)) / 0.3) * (1 / 0.3) + 20 * (1 / 0.3)]
Simplifying further:
dc/dp = 0.09 * [4((30 - ((30 - p) / 0.3)) / 0.3) * (1 / 0.3) + 20 * (1 / 0.3)]
dc/dp = 0.09 * [(4 / 0.3)((30 - ((30 - p) / 0.3)) * (1 / 0.3) + 20 * (1 / 0.3))]
Now, set the derivative equal to zero:
0.09 * [(4 / 0.3)((30 - ((30 - p) / 0.3)) * (1 / 0.3) + 20 * (1 / 0.3))] = 0
Solving this equation will give you the value of p that minimizes the cost function c.