3.4.1: Preparation 3.4
- Page ID
- 148746
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Use the formulas you found earlier to answer Questions 1 and 2.
(1) A courtyard shaped like a trapezoid is shown below. Find the area of the courtyard in ft2.
(2) Use the formula for the volume of a cylinder to find the volume of a water tank (in ft3) with a radius of 10 feet and a height of 40 feet. Round to two decimal places.
Subscripts
The following information will be used in Collaboration 3.4.
A subscript is a symbol that is written in small type below a variable in regular type. For example: P0 is read as P sub-zero.
Subscripts are used to distinguish between variables that represent similar quantities. For example, if you were working with a problem in which there were different prices over time, you might want to use P to represent all of those prices, but you also want to be able to tell the difference between them. So you could use P0 for the initial price, then P1 (P sub-one) for the second price. The subscript is only a label. It is not an operation.
The Grade of a Road
The grade of a road quantifies the rate of increase (or decrease) in height of the road over some horizontal span. The grade of a road is important information for drivers of large trucks in mountainous terrain. If a trucker begins to travel too fast going downhill, then it is possible for brakes to fail. Of course, as the driver of a car, you might be frustrated with a truck that is traveling up a hill with a steep grade, especially if you cannot pass. Runners and bicyclists who compete in hilly terrain also consider the grade of the hill in predicting their stamina and speed.
The grade of a road is written as a fraction with the numerator being the change in height (vertically) and the denominator being the change in distance (horizontally). In equations, the grade must be used as a positive value for cars moving uphill and as a negative value for cars moving downhill. (Note: when the grade of a road is reported, a sign is often not included because if one car is traveling uphill on a road, then oncoming cars are traveling downhill, so it could be confusing to report the grade as + or –.)
Notice that the units divide out when calculating grade, leaving a number that is dimensionless. That number is then written as a percent.
Example: Determine the grade of a road that decreases 72 feet in height over a horizontal distance of 600 feet.
Answer: \(\dfrac{-72\;feet}{600\;feet} = \dfrac{-72}{600} = \large{−0.12} = \large{−12\%}\)
After Preparation 3.4 (survey)
You should be able to do the following things for the next collaboration. Rate how confident you are on a scale of 1–5 (1 = not confident and 5 = very confident).
Before beginning Collaboration 3.4, you should understand the concepts and demonstrate the skills listed below.
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Skill or Concept: I can… |
Rating from 1 to 5 |
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evaluate expressions containing parentheses and exponents of two. |
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understand dimensional analysis and how to use units—including squared units—in calculations. |
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understand the use of subscripts. |