Question

The county administrator's office is making plans to finance the next phase of infrastructure repair to the county's roads and bridges. To help with their planning, they have developed functions that model the anticipated tax money available and planned budget for repairs, for any given year, x years from today.

The expected tax money, in millions, received from the state is represented by the function T(x) = 14x3 + 43x2 + 37x + 190. The planned budget allocating money, in millions, for infrastructure repairs in that same year is represented by the function R(x) = 35x + 90.

Which function represents the balance, in millions, that remains in a given year, after using a portion of the tax money to pay for infrastructure repairs?

Answer 1

Explanation:

The function that represents the balance, in millions, that remains in a given year after using a portion of the tax money to pay for infrastructure repairs is calculated by subtracting the planned budget for repairs from the expected tax money.

To find the balance function, we need to subtract the function representing the planned budget for repairs (R(x)) from the function representing the expected tax money (T(x)).

The function representing the balance, B(x), is given by:

B(x) = T(x) - R(x)

Substituting the given functions T(x) and R(x) into the equation, we have:

B(x) = (14x^3 + 43x^2 + 37x + 190) - (35x + 90)

Simplifying the equation by combining like terms, we get:

B(x) = 14x^3 + 43x^2 + 37x + 190 - 35x - 90

Further simplifying, we have:

B(x) = 14x^3 + 43x^2 + (37x - 35x) + (190 - 90)

Combining like terms again, we get:

B(x) = 14x^3 + 43x^2 + 2x + 100

Therefore, the function representing the balance, in millions, that remains in a given year after using a portion of the tax money to pay for infrastructure repairs is B(x) = 14x^3 + 43x^2 + 2x + 100.

This function represents the amount of money that remains after allocating the planned budget for repairs, taking into account the expected tax money. The balance function allows the county administrator's office to evaluate the financial situation and make informed decisions regarding the allocation of funds for infrastructure repairs in a given year.

Answer 2

Explanation:

The balance, in millions, that remains in a given year, after using a portion of the tax money to pay for infrastructure repairs can be represented by the function B(x), which can be calculated by subtracting the planned budget (R(x)) from the expected tax money (T(x)):

B(x) = T(x) - R(x)

Given:

T(x) = 14x^3 + 43x^2 + 37x + 190 (expected tax money)

R(x) = 35x + 90 (planned budget)

To Find:

Function B(x) representing the balance remaining after allocating a portion of tax money for repairs.

Solution:

Substitute the given functions T(x) and R(x) into the equation for B(x):

B(x) = T(x) - R(x)

B(x) = (14x^3 + 43x^2 + 37x + 190) - (35x + 90)

Simplify the equation:

B(x) = 14x^3 + 43x^2 + 37x + 190 - 35x - 90

B(x) = 14x^3 + 43x^2 + 2x + 100

Final Answer:

The function representing the balance, in millions, that remains in a given year after using a portion of the tax money to pay for infrastructure repairs is B(x) = 14x^3 + 43x^2 + 2x + 100.