The marginal cost for a product is given by dC/dq = 1000/sqrt3q+20, in hundreds of dollars, where q is given in thousands of units. Determine the cost in dollars involved in increasing product from 10 to 15 thousand units. If

R(0)= C(0)= 0, the marginal revenue is given by dR/dq =30(2-0.1q) and the surplus equation given by Ps(q)= 10 + 18.5q, determine the consumers’ surplus if the unit price is equal to the market value when in equilibrium.

To determine the cost involved in increasing the product from 10 to 15 thousand units, we need to calculate the integral of the marginal cost function over the range of q from 10 to 15.

The given marginal cost function is dC/dq = 1000/sqrt(3q+20).

Here's how you can calculate the cost involved:

1. Integrate the marginal cost function with respect to q:
C(q) = ∫(1000/√(3q+20))dq

2. Evaluate the integral by finding the antiderivative of the function:
C(q) = 1000 ∫(3q+20)^(-1/2)dq

3. Apply the power rule of integration:
C(q) = 1000 [2(3q+20)^(1/2)]

4. Evaluate the integral at the upper and lower limits:
C(15) - C(10) = 1000 [2(3(15)+20)^(1/2) - 2(3(10)+20)^(1/2)]

5. Calculate the difference:
C(15) - C(10) = 1000 [2(45+20)^(1/2) - 2(30+20)^(1/2)]

6. Simplify the expression within the square root:
C(15) - C(10) = 1000 [2(65)^(1/2) - 2(50)^(1/2)]

7. Calculate the value within the square root:
C(15) - C(10) = 1000 [2(8.06) - 2(7.07)]

8. Simplify the expression:
C(15) - C(10) = 1000 [16.12 - 14.14]

9. Calculate the difference:
C(15) - C(10) = 1000 [1.98]

10. Multiply by 100 to convert the cost to dollars:
C(15) - C(10) = $1980

Therefore, the cost involved in increasing the product from 10 to 15 thousand units is $1980.

To determine the consumer surplus, we need to find the area between the demand curve (marginal revenue) and the market price curve (surplus equation).

1. The marginal revenue is given by dR/dq = 30(2 - 0.1q).

2. Calculate the integral of the marginal revenue function from 0 to Q, where Q is the quantity in equilibrium:
R(Q) = ∫(30(2 - 0.1q))dq

3. Evaluate the integral by finding the antiderivative of the function:
R(Q) = 30 [ 2q - 0.05q^2 ]

4. Calculate the integral at the upper limit:
R(Q) = 30 [ 2Q - 0.05Q^2 ]

5. Substitute the value of Q where the unit price is equal to the market value in equilibrium into the surplus equation, Ps(q) = 10 + 18.5q:
Ps(Q) = 10 + 18.5Q

6. Calculate the consumer surplus by finding the difference between the revenue and the surplus at equilibrium:
Consumer Surplus = R(Q) - Ps(Q)

7. Substitute the expressions calculated above into the consumer surplus equation and simplify the expression:
Consumer Surplus = 30 [ 2Q - 0.05Q^2 ] - (10 + 18.5Q)

8. Simplify the expression:
Consumer Surplus = 60Q - 1.5Q^2 - 10 - 18.5Q

9. Combine like terms:
Consumer Surplus = -1.5Q^2 + 41.5Q - 10

Therefore, the consumer surplus if the unit price is equal to the market value when in equilibrium is given by the equation -1.5Q^2 + 41.5Q - 10.