Consider the following problem:
The horizontal leg $a(t)$ of a right triangle is increasing at a rate of $8 \mathrm{~m} / \mathrm{s}$ and the vertical leg $b(t)$ of the triangle is increasing at a rate of $7 \mathrm{~m} / \mathrm{s}$. At a certain instant $t_{0}$, the horizontal leg is $13 \mathrm{~m}$ and the vertical leg is $10 \mathrm{~m}$. What is the rate of change of the length $c(t)$ of the hypotenuse at that instant?

Match each expression with its units.

Question

Consider the following problem:
The horizontal leg $a(t)$ of a right triangle is increasing at a rate of $8 \mathrm{~m} / \mathrm{s}$ and the vertical leg $b(t)$ of the triangle is increasing at a rate of $7 \mathrm{~m} / \mathrm{s}$. At a certain instant $t_{0}$, the horizontal leg is $13 \mathrm{~m}$ and the vertical leg is $10 \mathrm{~m}$. What is the rate of change of the length $c(t)$ of the hypotenuse at that instant?

Match each expression with its units.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We start by noting the given information: $a(t) = 13 \, ext{m}$, $b(t) = 10 \, ext{m}$, $\frac{da}{dt} = 8 \, ext{m/s}$, and $\frac{db}{dt} = 7 \, ext{m/s}$. We need to find $\frac{dc}{dt}$, where $c(t)$ is the hypotenuse of the right triangle. ‖ ‖ step 2 ⋮ Using the Pythagorean theorem, we have $c(t) = \sqrt{a(t)^2 + b(t)^2}$. ‖ ‖ step 3 ⋮ Differentiating both sides with respect to $t$, we get $\frac{d}{dt}[c(t)] = \frac{d}{dt}[\sqrt{a(t)^2 + b(t)^2}]$. ‖ ‖ step 4 ⋮ Applying the chain rule, we obtain $\frac{dc}{dt} = \frac{1}{2\sqrt{a(t)^2 + b(t)^2}} \cdot 2a(t)\frac{da}{dt} + 2b(t)\frac{db}{dt}$. ‖ ‖ step 5 ⋮ Simplifying, we get $\frac{dc}{dt} = \frac{a(t)\frac{da}{dt} + b(t)\frac{db}{dt}}{\sqrt{a(t)^2 + b(t)^2}}$. ‖ ‖ step 6 ⋮ Substituting the given values, we have $\frac{dc}{dt} = \frac{13 \cdot 8 + 10 \cdot 7}{\sqrt{13^2 + 10^2}} = \frac{104 + 70}{\sqrt{169 + 100}} = \frac{174}{\sqrt{269}}$. ‖ ‖ step 7 ⋮ Simplifying further, $\frac{dc}{dt} = \frac{174}{\sqrt{269}} \approx 10.6 \, ext{m/s}$. ‖ ∻Answer∻ ⚹ The rate of change of the length of the hypotenuse is approximately $10.6 \, ext{m/s}$. ⚹ ∻Key Concept∻ ⚹Rate of change of the hypotenuse in a right triangle⚹ ∻Explanation∻ ⚹The rate of change of the hypotenuse can be found by differentiating the Pythagorean theorem with respect to time and substituting the given rates of change of the legs.⚹◊What is the formula for finding the rate of change of the hypotenuse $c(t)$ in a right triangle at a certain instant $t_{0}$?⍭ Generate me a similar question◊

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