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$$f(x)=5x^3-8x^2+1$$ There is a root between x=1 and x=2, use linear interpolation (using similar triangles) to find the root correct to 1 dp

So I tried doing this: $$f(1)=-2$$ $$f(2)=9$$

so the ratio ought to be $$(a-1)/(2-a)=2/9$$ which you then rearrange to find $a$ however the answer book says it is $$(a-1)/(2-a)=9/2$$ which I can't understand.

I know it should be done with similar triangles but I don't see where their ratio comes from.

user324496
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1 Answers1

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You have three points on your line: $(1,-2)$, $(2,9)$ and your root $(a, 0)$.

Using the fact that the slope between any two of them is the same we get that the rise over run going from $(a,0)$ to $(2,9)$ is $\frac{9}{2-a}$.

The rise over run from $(1, -2)$ to $(a, 0)$ is $\frac{2}{a-1}$.

Equating the two ratios gives $\frac{9}{2-a}=\frac{2}{a-1}\implies \frac{(a-1)}{(2-a)}=\frac{2}{9}$.

Therefore, you are correct.

John Douma
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