Health Expenditures of the functionH(x) = 0.004x^2 - 0.197x + 5.406 models the percentage of total income that the individual who is x years old spends on health care.  a) use the model to...

Health Expenditures of the function

H(x) = 0.004x^2 - 0.197x + 5.406

models the percentage of total income that the individual who is x years old spends on health care. 

a) use the model to approximate the percentage of total income an individual who is x=45 years old spends on healthcare. 

0.004(45)^2 - 0.197(45) + 5.406=

0.004(2025) - 0.197(45) + 5.406

8.1 - 8.865 + 5.406 = 4.641 %

I'm not understanding how to figure part B of this question.

b) At what age is the percentage of income spent on health care 10%?

Asked on by deekoch

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

In the part a), you've found out the total income spent on health by an individual who is 45 years old. Therefore, you have substituted x by the age of the individual and you have determined the value of the function H(x). This value represents the percentage of total income that the individual spends on health care. 

In the part b), you know the value of the function H(x) = 10%, but you don't know the age, therefore, you'll have to solve a quadratic equation to find the ages x.

`0.004x^2 - 0.197x + 5.406 = 10%`

Write `10% = 10/100 = 1/10 = 0.1`

`0.004x^2 - 0.197x + 5.406 = 0.1`

Subtract 0.1 both sides:

`0.004x^2 - 0.197x + 5.406 -0.1 = 0`

`` Use quadratic formula to find x.

`x_(1,2) = (0.197 +- sqrt (0.038809-0.021224))/0.008`

`x_(1,2) = (0.197 +- 0.132)/0.008`

`x_1` =41 years old

`x_2` =8 years old

ANSWER: At the age of 41 years old, an individual spends on health care 10% of total income. An individual of 8 years old may also spend on health care 10% of total income.

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