A builder makes drainpipes that drop $1 \mathrm{~cm}$ over a horizontal distance of $30 \mathrm{~cm}$ to prevent clogs. See the diagram below, which is not drawn to scale:

A certain drainpipe needs to cover a horizontal distance of $700 \mathrm{~cm}$.

What is the length $\ell$ of this drainpipe?
Round your answer to the nearest tenth of a centimeter.
$\square$ $\mathrm{cm}$

Question

A builder makes drainpipes that drop $1 \mathrm{~cm}$ over a horizontal distance of $30 \mathrm{~cm}$ to prevent clogs. See the diagram below, which is not drawn to scale:

A certain drainpipe needs to cover a horizontal distance of $700 \mathrm{~cm}$.

What is the length $\ell$ of this drainpipe?
Round your answer to the nearest tenth of a centimeter.
$\square$ $\mathrm{cm}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the length $\ell$ of the drainpipe, we use the similarity of the two right triangles. The triangles are similar because they both have a right angle and they share the angle where the hypotenuse and the horizontal side meet, making the third angles equal by the Angle-Angle Similarity Postulate. ‖ ‖ step 2 ⋮ The ratio of the vertical side to the horizontal side for the smaller triangle is $\frac{1}{30}$. This ratio will be the same for the larger triangle because of the similarity. ‖ ‖ step 3 ⋮ Set up a proportion using the corresponding sides of the similar triangles: $\frac{1}{30} = \frac{x}{700}$, where $x$ is the vertical side of the larger triangle. ‖ ‖ step 4 ⋮ Solve the proportion for $x$: $x = \frac{700}{30} = \frac{70}{3}$. ‖ ‖ step 5 ⋮ Use the Pythagorean theorem for the larger triangle: $700^2 + x^2 = \ell^2$. ‖ ‖ step 6 ⋮ Substitute $x = \frac{70}{3}$ into the equation: $700^2 + \left(\frac{70}{3}\right)^2 = \ell^2$. ‖ ‖ step 7 ⋮ Calculate $\ell^2$: $700^2 + \left(\frac{70}{3}\right)^2 = 490000 + \frac{4900}{9} = \ell^2$. ‖ ‖ step 8 ⋮ Simplify and solve for $\ell$: $\ell = \sqrt{490000 + \frac{4900}{9}}$. ‖ ‖ step 9 ⋮ Calculate the value of $\ell$ and round to the nearest tenth: $\ell \approx \sqrt{490000 + 544.44} \approx \sqrt{490544.44} \approx 700.4$. ‖ ∻[1] Answer∻ ⚹ $\ell \approx 700.4$ cm ⚹ ∻Key Concept∻ ⚹Similar Triangles and the Pythagorean Theorem⚹ ∻Explanation∻ ⚹The length of the drainpipe is found by using the properties of similar triangles to determine the length of the missing side and then applying the Pythagorean theorem to find the hypotenuse, which is the length of the drainpipe. The answer is rounded to the nearest tenth as required.⚹◊What is the formula for finding the hypotenuse of a right triangle when given the lengths of the other two sides?⍭ Generate me a similar question◊

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