8.2E Exercises

Last updated
Save as PDF

Page ID: 157061

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Fences

Cretaceous Park has 1200m of fencing material to build a paddock for their velociraptors (velociraptors are from the late Cretaceous period, not the Jurassic).

If they want to make a square paddock, what should its dimensions be? What would be the resulting area inside the fencing?
If they want to make a rectangular paddock whose long side is twice the short side, what should the dimensions be? What would be the area in this case?
Velociraptors are scared of fast water (I made this up, don't fact-check me), so the park can use a straight-running river on his property as the "fourth wall" of the paddock, thus only using fencing for three sides. If he wants a square paddock again, still using up all of the fencing material, what should the dimensions be? What would be the area?

Answer

300m by 300m. Area will be 90,000 square meters.
200m by 400m. Area will be 80,000 square meters.
400m by 400m. Area will be 160,000 square meters.

$paddocks.png$

Note: You can use a calculator on the rest of these problems.

Airplanes

Plane A takes off from an airport and flies due west at a speed of 550mph. At the same time, Plane B leaves the same airport and flies due south at a speed of 570mph.

After 2 hours, how far has Plane A traveled from the airport? How far as Plane B traveled?
After 2 hours, how far apart (as the crow flies) are Plane A and Plane B?
After a certain amount of time, Plane A and Plane B are 990mi apart, and Plane A is 687.5mi west of the airport. How far is Plane B from the airport?

Answer

Plane A has gone 1100mi. Plane B has gone 1140mi.
The distance between them is about 1584mi.
This time the hypotenuse and one leg of the triangle are given. The length of the other leg is 712.5mi.

$airport.png$

Crossing a River

Town A is on the bank of a 40-meter-wide river, and Town B is on the other bank, 1 kilometer downstream. I start in Town A and run 500 meters down the bank. Then I jump into the river and swim directly towards Town B, diagonally across the water.

How far do I have to swim? How much distance do I cover total, both running and swimming?
If I were to jump directly in the river at Town A and swim diagonally through the river to Town B, how far would I swim?
If I were to run 700m first and then start swimming, how far would I swim? What is the total distance traveling from Town A to Town B?

Answer

The swim is about 501.6 m. The total distance is then 1001.6 m.
The swim is about 1000.8 m, and that's the full distance.
The swim is about 302.7 m. The total distance covered is 1002.7 m.

Figure is not drawn to scale!

$river.png$

Building an Aquarium

I'm designing a tank for my axolotl and I will have to order pieces of plate glass to assemble it out of. The plate glass will cost me $25 per square foot. I will use a different material to make the lid of my aquarium, so I am going to build an open glass box.

If I want the tank to be 2ft wide, 3ft long, and 2ft tall, how much will the glass cost me?
If I want to make a cubical tank and can't afford more than $600 worth of glass, what's the biggest size tank I could make (find the max side length)?

Answer

It will cost me $650.
The side length should be about 2.2 ft.

Conical Tank

Express the radius of the circular water surface, $r$, in terms of the height of the water level, $h$. If the height increases by 0.5ft, how will the radius $r$ change? Express the volume of water inside the tank as a function of $h$.

$cone1.png$

Answer

Using similar triangles, we have $ \dfrac{r}{1} = \dfrac{h}{1}$... oh wait, $r = h$ here. Did you recognize that the triangles are 45-45-90? The leg lengths are equal. If height changes by any amount, then since $r = h$, the radius will change by the same amount!

The volume of a cone is $V = \frac{1}{3} \pi r^2 h $, and we use the fact that $r = h$ in this case to express $V(h) = \frac{1}{3} \pi h^3$ as a function of $h$ alone.

Water Trough

A trough, shown below, is filled partway with water.

Find the maximum volume capacity of the trough.
Express the dimension $w$ in terms of the water level height $h$.
Write the volume of the water as a function of $h$ alone.

$trough.png$

Answer

8 cubic feet.
Using similar triangles, $ \dfrac{w}{2} = \dfrac{h}{2}$ so $w = h$.
$V(h) = 2wh = 2h^2$.

Distance Formula

Find the distance between the points.

$(1,2)$ and $ (3,4)$
$ (-1, -1) $ and $ (0, 5) $
$ (2, 7)$ and $ (0,0)$
$ (10,2)$ and $ (-4, 5)$

Answer

$\sqrt{8} \approx 2.83$
$ \sqrt{37} \approx 6.08 $
$ \sqrt{53} \approx 7.28 $
$ \sqrt{205} \approx 14.32 $

Search

Text Color

Text Size

Margin Size

Font Type