Quiz 7 (Solution)


Date: MATH 1210-4 - Spring 2004

Note: Unless stated otherwise, answers without justification receive no credit. Please, show your work!

  1. Use differentials to approximate $ \sqrt{62}$.

    Since $ 62$ is close to $ 64$, whose square root is $ 8$, we use the formula

    $\displaystyle y(x_0 + \Delta x) \simeq y(x_0)+ y'(x_0) \Delta x,$    

    for $ y(x)=\sqrt{x}$, $ x_0=64$, and $ \Delta x = -2$. Then, since $ y'(x)
= \frac{1}{2\sqrt{x}} \ensuremath{\Rightarrow}\xspace y'(64) = \frac{1}{2\sqrt{64}} =
\frac{1}{16}$ we have

    $\displaystyle \sqrt{62} = y(64-2) \simeq y(64) + y'(64) (-2) = 8 - \frac{2}{16} = \frac{63}{8} = 7.875.$    

    Using the calculator we obtain the value (rounded to five decimal places) $ 7.87401$.

  2. An airplane, flying horizontally, at an altitude of 1 mile, passes directly over an observer. If the constant speed of the airplane is 600 miles per hour, how fast is its distance from the observer increasing 1 minute later?

    Figure 1: Schematic description for the airplane problem
    \includegraphics[scale=.5,angle=0,
clip= true]{q7-f0.eps}

    Figure 1 shows the graph where $ s$ is the distance from the airplane to the observer and $ x$ is the (horizontal) distance traveled by the airplane from the moment it passed over the observer.

    We know that $ \frac{dx}{dt} = 600$ and we want to know $ \frac{ds}{dt}$ one minute after the plane flew over the observer. Since the plane travels $ \frac{600}{60} = 10$ miles per minute, we want to know $ \frac{ds}{dt}$ when $ x=10$.

    For all times we have the relation $ s^2 = 1^2 + x^2$, so that, taking derivatives (with respect to time, $ t$) on both sides we get

    $\displaystyle 2s\frac{ds}{dt} = 2x\frac{dx}{dt},$    

    so that

    $\displaystyle \frac{ds}{dt} = \frac{x}{s} \frac{dx}{dt}.$    

    Now we see that when $ x=10$, $ s=\sqrt{1^2+10^2} = \sqrt{101} \simeq
10.049876$, and we obtain

    $\displaystyle \frac{ds}{dt} = \frac{x}{s} \frac{dx}{dt} = \frac{10}{10.049876} \cdot 600 = 597.02231.$    



Javier Fernandez 2004-03-22