How would i start this problem?
The amount that workers contribute monthly for health care insurance premiums can be modeled by A(t)=0.07t^3-3.1t^2 + 54.3t-230, where A is the monthly amount contributed and t is the number of years after 1980. Find the instantaneous rate of change in monthly contribution in 2000.
Ah, the joys of math problems! Let's get started on this one.
To find the instantaneous rate of change in the monthly contribution in 2000, we need to find the derivative of the function A(t). In this case, we will be finding the derivative with respect to t.
So, if we differentiate A(t) = 0.07t^3 - 3.1t^2 + 54.3t - 230 with respect to t, we get:
A'(t) = d/dt (0.07t^3 - 3.1t^2 + 54.3t - 230)
But hold on, let's not forget that t is the number of years after 1980, so we need to adjust that. Since we are looking for the rate of change in 2000, we need to find A'(20) since 2000 is 20 years after 1980.
Now, let's input t = 20 into our derivative:
A'(20) = d/dt (0.07(20)^3 - 3.1(20)^2 + 54.3(20) - 230)
And after evaluating that expression, you'll have your answer! Voila!
To find the instantaneous rate of change in monthly contribution in 2000, you need to find the derivative of the function A(t) and then evaluate it at t = 2000.
Step 1: Take the derivative of A(t)
To find the derivative of A(t), you can use the power rule for differentiation. The power rule states that if you have a term of the form ax^n, the derivative is given by nx^(n-1). Applying this rule to each term in A(t), we get:
A'(t) = d/dt (0.07t^3) - d/dt (3.1t^2) + d/dt (54.3t) - d/dt (230)
The derivatives of the individual terms are:
d/dt (0.07t^3) = 0.21t^2
d/dt (3.1t^2) = 6.2t
d/dt (54.3t) = 54.3
d/dt (230) = 0
Therefore, the derivative of A(t) is:
A'(t) = 0.21t^2 - 6.2t + 54.3
Step 2: Evaluate the derivative at t = 2000
To find the instantaneous rate of change in monthly contribution in 2000, substitute t = 2000 into the derivative A'(t):
A'(2000) = 0.21(2000)^2 - 6.2(2000) + 54.3
Now, you can simplify this expression and calculate the value to find the instantaneous rate of change in monthly contribution in 2000.