Question

New underwater camera and sonar accessories are going to be installed on the fishing boat to detect the movement of mackerel fish up to 133 ft water depth. It is estimated that the present worth now of income from the new installation is $13,500,000. There will be no income in years 1 to 3, but in year 4 income will be $200,000, and the amount is arithmetically increased through year 15. If the interest rate is 12% per year, find the required gradient

Answer 1

To find the required gradient, calculate the present worth of income for each year and solve the equation to find g.

Step-by-step explanation:

To find the required gradient, we need to calculate the present worth of income for each year from 4 to 15, and then find the constant gradient that would result in a present worth of $13,500,000. The present worth of income for each year can be calculated using the formula for present worth:

P = A / (1 + r)^n

where P is the present worth, A is the future value (income in this case), r is the interest rate, and n is the number of years.

Substituting the values into the formula, we get:

P4 = $200,000 / (1 + 0.12)^4 = $122,386.28

P5 = P4 * (1 + g) = $122,386.28 * (1 + g)

P15 = P4 * (1 + g)^11 = $122,386.28 * (1 + g)^11

Setting the sum of the present worths equal to $13,500,000:

P4 + P5 + ... + P15 = $13,500,000

$122,386.28 + ($122,386.28 * (1 + g)) + ... + ($122,386.28 * (1 + g)^11) = $13,500,000

This equation can be solved for the required gradient, g, using algebraic techniques such as factoring or substitution.