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zysi [14]
3 years ago
6

A house appreciates (grows in value) about 6% each year. If the house is now worth 158,000, in about how many years will it be w

orth 200,000? Use the equation 200,000= 158,000(1.06)^x to solve this problem. Round to the nearest year
Mathematics
1 answer:
expeople1 [14]3 years ago
8 0
200,000 = 158,000(1.06)^x
Divide both sides by 158,000
1.2658 = 1.06^x
Take the log of both sides and use the power rule to bring down the exponent (x) to the front.
log 1.2658 = x log 1.06
Divide both sides by log 1.06
log (1.2658)/log(1.06) = x
x = 4 years

CHECK
y = 158,000(1.06)^4
y = 199,471.36
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algol [13]

We need to differentiate this with respect to x to see if we can find an expression for the derivative of y at various points.  That will be the slope of the tangent to the curve.  Then we want to see where that derivative might be infinite -- i.e., where the tangent is vertical.

 

It's not written as a function, but it can still be differentiated using the chain rule:

 

x2 + xy + y2 = 3

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(I used parentheses to show the differentiation of each term in the original equation.)

 

2x + x dy/dx + y + 2y dy/dx = 0

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We have the derivative of y, but it's defined partly in terms of y itself.  That's OK.  Let's go on...

 

So where would the slope be infinite?  That would happen when x + 2y = 0, or y = -x/2

 

Let's plug that in for y in the original equation to find points where that's the case.

 

x2 + xy + y2 = 3

x2 + x(-x/2) + (-x/2)2 = 3

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So at the point (2, -1) the tangent is vertical.

 

Now try x = -2...

 

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Answer:

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Finally, you can double check your math.

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