Pre calculus word problem

Question

After declining between 1940 and 1980, the number of multigenerational households has been increasing since 1980. The function h(x)= 0.012x^2-0.573x+35.722 can be used to estimate the number of multigenerational households in country A, in millions, x years after 1940. In what year were there 50 million multigenerational households? There were 50 million multigenerational households in? (Round the nearest whole number)

Answer 1

Hello again!

For this problem, we know that h(x) = 0.012x² - 0.573x + 35.722 is the number of households (in millions) x years after 1940. (So if it was 1950 for example, x = 10 since 1950 is 10 years after 1940).

We want to know, in what year were there 50 million households? In other words, since we want to find the year, we need to figure out what x equals.

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How can we do that? Well, we know that h = 50 (that's 50 million households), so I'll write:

50 = 0.012x² - 0.573x + 35.722.

To solve this quadratic equation, we'll set equal to 0 and use the quadratic formula:

0 = 0.012x² - 0.573x - 14.278.

The coefficients are: a = 0.012, b = -0.573, and c = -14.278.

The quadratic formula is:

So, plugging in our coefficients we get:

and after some simplifying the possible answers are: x = 65.826 or x = -18.076. The negative answer for x doesn't make sense (-18 years after is confusing), so the one we want must be x = 65.826.

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So, in what year were there 50 million households? Well, since x = 65.826, that must mean that the year was about 66 years after 1940. This would be 1940 + 66 = 2006.

Hope that helps! Let me know if anything was unclear. :)