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Solve one-step multiplication and division equations: word problems

Anatoli was given this problem:\newlineTwo cars are driving towards an intersection from perpendicular directions. The first car's velocity is 22 meters per second and the second car's velocity is 99 meters per second. At a certain instant t0 t_{0} , the first car is a distance x(t0) x\left(t_{0}\right) of 88 meters from the intersection and the second car is a distance y(t0) y\left(t_{0}\right) of 66 meters from the intersection. What is the rate of change of the distance d(t) d(t) between the cars at that instant?\newlineWhich equation should Anatoli use to solve the problem?\newlineChoose 11 answer:\newline(A) tan⁡[d(t)]=y(t)x(t) \tan [d(t)]=\frac{y(t)}{x(t)} \newline(B) d(t)=x(t)⋅y(t)2 d(t)=\frac{x(t) \cdot y(t)}{2} \newline(C) d(t)+x(t)+y(t)=180 d(t)+x(t)+y(t)=180 \newline(D) [d(t)]2=[x(t)]2+[y(t)]2 [d(t)]^{2}=[x(t)]^{2}+[y(t)]^{2}
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Lea was given this problem:\newlineA 2020-meter ladder is sliding down a vertical wall so the distance y(t) y(t) between the top of the ladder and the ground is decreasing at 88 meters per minute. At a certain instant t0 t_{0} , the top of the ladder is 1212 meters from the ground. What is the rate of change of the distance x(t) x(t) between the bottom of the ladder and the wall at that instant?\newlineWhich equation should Lea use to solve the problem?\newlineChoose 11 answer:\newline(A) sin⁡[20]=y(t)x(t) \sin [20]=\frac{y(t)}{x(t)} \newline(B) 20=x(t)⋅y(t)2 20=\frac{x(t) \cdot y(t)}{2} \newline(C) 20+x(t)+y(t)=180 20+x(t)+y(t)=180 \newline(D) 202=[x(t)]2+[y(t)]2 20^{2}=[x(t)]^{2}+[y(t)]^{2}
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Let f(x)=2x−sin⁡(πx) f(x)=2^{x}-\sin (\pi x) .\newlineBelow is Rafael's attempt to write a formal justification for the fact that the equation f′(x)=14 f^{\prime}(x)=\frac{1}{4} has a solution where −2<x<−1 -2<x<-1 .\newlineIs Rafael's justification complete? If not, why?\newlineRafael's justification:\newlineExponential and trigonometric functions are differentiable and continuous at all points in their domain, and −2≤x≤−1 -2 \leq x \leq-1 is within f f 's domain.\newlineSo, according to the mean value theorem, f′(x)=14 f^{\prime}(x)=\frac{1}{4} must have a solution somewhere in the interval\newline−2<x<−1.  -2<x<-1 \text {. } \newlineChoose 11 answer:\newline(A) Yes, Rafael's justification is complete.\newline(B) No, Rafael didn't establish that the average rate of change of f f over [−2,−1] [-2,-1] is equal to 14 \frac{1}{4} .\newline(C) No, Rafael didn't establish that f f is differentiable.
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