3.3.1: Preparation 3.3
- Page ID
- 148743
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The next collaboration explores calculations needed for repairs and improvements to a house and lot. The problems will require that you understand concepts of length, area, and volume and appropriate units of measure, based on customary U.S. units.
Length
Length is one-dimensional. In the context of remodeling a house, an example would be the total length of baseboard needed to trim the walls of a room. Examples of units of measure for length are inches, feet, yards, or miles. A number line can be used to model lengths. The thicker segment on each number line is 3 units long. If the scale is in inches, each line segment is 3 inches long. If the scale is in feet, each line segment is 3 feet long.
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A Number Line |
Area
Area is two-dimensional and is measured in square units. The total number of one-foot square tiles needed to cover the floor of a room is an example of area measured in square feet. A rectangle is one shape that can be used to model area. Recall the formula for the area of a rectangle: A = L × W. The area of a rectangle is the product of the length and the width, which is a shortcut for counting the number of square units needed to cover the rectangle.
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A Coordinate Axis |
Each of the two shaded areas on the coordinate axis has an area of 12 square units. If the horizontal and vertical scales are in inches, each area is 12 square inches. If the scales are in feet, each area is 12 square feet. Notice that the regions measured do not have to be squares, yet the area is measured in square units.
Notice how the units in the calculation determine the units in the result:
A = (2 inches) × (6 inches)
(2 × 6) (inches × inches)
12 square inches or inches2
If the units are feet, the area of the rectangle on the top is A = (3 feet) × (4 feet) = 12 square feet or feet2.
Note 1: It is common to abbreviate the units of measure using exponents. Since the area might be 12 feet × feet, write A = 12 ft2. Notice the connection to algebra here! Multiplying (3 feet) by (4 feet) is similar to multiplying (3x) by (4x). You multiply the numbers in front of the variables (coefficients), and then multiply the variables: (3 ⋅ 4) (x ⋅ x) = 12x2.
Note 2: It is common to confuse length and area formulas. To calculate the length of the line surrounding the rectangle, which is called the perimeter, simply add the total number of units as if traveling around the area. For example, if the units are in feet, then the length of the line around the bottom rectangle is 2 feet + 6 feet + 2 feet + 6 feet = 16 feet. The arithmetic operation is addition, and the unit of measure is feet. By comparison, the arithmetic operation to compute area is multiplication and the unit of measure is square feet. Again, this connects to algebra. To add algebraic terms, you must have like terms, meaning terms with the same variables: 2x + 6x + 2x + 6x = 16x. You cannot add 2x + 3y just as you cannot add 2 feet + 3 inches.
The area of the circle is given by the formula, A = πr2.
- π is a constant that is approximately 3.14159 (you probably learned 3.14, but that can lead to rounding errors).
- r is the radius of the circle, which varies. (A radius of a circle is the distance from the center of a circle to its outside edge).
- A is the area of the circle, which varies.
(1) Write each of the following products (the result to a multiplication problem) using exponents to express the results in a simpler form.
(a) (3a)(5a)
(b) (5p)(2p)
(c) (3 inches)(5 inches)
(d) (5 feet)(2 feet)
(2) How many square inches are in 1 square foot?
(3) The formulas for finding the area of two-dimensional geometric figures that occur in everyday use are published in reference books or available online. Use the Internet or some other reliable source to find a formula for the area of each figure. Define each variable in the formula, and label the figure with the variables to indicate the correct meaning of the variable. You may have to add to the figure to indicate all variables.
Example: Rectangle
Variables: L = length; W = width
Area of a rectangle = L × W
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(a) Parallelogram Variables: Area of parallelogram = |
(b) Triangle Variables: Area of triangle = |
(c) Trapezoid Variables: Area of trapezoid = |
(4) Find the unknown quantity in each situation.
(a) A rectangle with an area of 72 square inches has a length of 6 inches. What is the width of the rectangle?
(b) A rectangle has a length of 9 inches and a width of 1.5 feet.
(i) What is the area of the rectangle in square inches?
(ii) What is the area of the rectangle in square feet?
(c) A rectangle with an area of 1 square foot has a width of 6 inches. What is the length of the rectangle?
(5) What is the area of a circle with a diameter of 5 feet? Round to the nearest tenth.
(6) If the radius of a circle is doubled, will the area also double? Explain your reasoning.
Hint: Compare the areas of two circles to see if it supports your reasoning: one circle with a radius of 10 inches and the other with a radius of 20 inches.
Volume
Volume is three-dimensional and is measured in cubic units. The formula to calculate the volume of a box is V = L ⋅ W ⋅ H. In the image below, the shaded volume is 3 units wide, 4 units tall, and 5 units long, a volume of 60 cubic units. A cubic unit is a unit that is cubed, or raised to the power of 3. If the scales are measured in inches, then the volume is V = (5 inches) ⋅ (3 inches) ⋅ (4 inches) = 5⋅3⋅4 cubic inches, or 60 in3. If the scales are in feet, the volume is 60 ft3 or 60 cubic feet.
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Note: The power of 3 is often called the cube of a number just as the power of 2 is called the square of a number. So 53 can be called 5 cubed.
(7) A cubic foot is a cube that is one foot on each side. How many cubic inches are in one cubic foot?
Roots
Sometimes we know the area of a square, and need to find the length of the sides. The formula for a square's area is A=L×L or A=L2. We have two choices for finding L: we can guess and check, revising our guesses to make them increasingly accurate, or we can use the square root function on a calculator. For example, a square with an area of 36 square inches has the equation 36 in2 = L2. The square root function (√36) tells us the exact value of L. In this case, L = 6 in. Generally, square roots are not whole numbers, but have decimals that continue forever without repeating. It is also true that (-6) × (-6) = 36, so there are actually two answers for every square. However, the negative result rarely applies to the context of the problem, and it is usually discarded.
It is also possible to find the length of the sides of a cube using the cube root function. For example, a cube with a volume of 8 ft3 would have a side that is 2 ft long because 8 = 23. Finding cube roots is typically beyond the scope of this course.
After Preparation 3.3 (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 3.3, 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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use basic formulas for area and volume (area of a rectangle and circle; volume of a box). |
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use appropriate units in calculations for length, area, and volume. |