Math word problem involving quadratic equations?

Question

An insurance comapany charges a 35 year old nonsmoker an annual preminum of $118 for a $100,000 term life insurance policy. The preminums for a 45 year old and 55 year old nonsmokers are $218 and $563, respectively. Write a quadratic modle for the preminum p as a function of age a.

Could you explain too?

Answer 1

If you're assuming that this is a quadratic relationship, then it should follow p = am^2 + bm + c (I used m for the age instead of a to avoid confusion with the constant a - we can change it back to a later when we're done). The problem here is that we don't know what the constants a, b, and c are, but we have some points that we can use and solve for them. If you plug in the numbers, you will have the following:

For a 35 year old's $118 premium,
118 = a(35^2) + b(35) + c
118 = 1225a + 35b + c

For a 45 year old's $218 premium,
218 = a(45^2) + b(45) + c
218 = 2025a + 45b + c

For a 55 year old's $563 premium,
563 = a(55^2) + b(55) + c
563 = 3025a + 55b + c

Now you have three equations and three unknowns. You can solve this using matrix operations, but in case you don't know how to do all of that, I'll solve it using the equations themselves.

If you take the second equation and subtract the first from it, you will have:

218 = 2025a + 45b + c
118 = 1225a + 35b + c
------------------------------
100 = 800a + 10b

Likewise for the third equation minus the second:

563 = 3025a + 55b + c
218 = 2025a + 45b + c
------------------------------
345 = 1000a + 10b

Now let's subtract the new first equation from the new second equation:

345 = 1000a + 10b
100 = 800a + 10b
------------------------------
245 = 200a
a = 1.225

We can start substituting back in to our equations to find the coefficients b and c:

100 = 800a + 10b
100 = 800(1.225) + 10b
100 = 980 + 10b
-880 = 10b
b = -88

118 = 1225a + 35b + c
118 = 1225(1.225) + 35(-88) + c
118 = 1500.625 - 3080 + c
118 = -1579.375 + c
c = 1697.375

So then your equation becomes:

p = 1.225a^2 - 88a + 1697.375

You can double check if this works by plugging in the original values into it - if you plug in 35 for a, you should get $118.

Answer 2

try 3. 3 always works