3.7.2: Exercise 3.7

MAKING CONNECTIONS TO THE COLLABORATION

(1) Which of the following was one of the main mathematical ideas of the collaboration?

(i) Graphic artists have to be very aware of proportionality and know how to solve proportions.

(ii) The rules for solving equations are the same for all types of equations.

(iii) The rules for solving equations depend on the type of equation.

(iv) To solve for a variable in the denominator of a fraction, multiply both sides of the equation by the variable.

(2) In Module 1, you learned that a statement such as “30% of voters support Candidate A” can be interpreted as 30 out of 100.

(a) How many voters out of 1,000 support Candidate A?

(b) How many voters out of 1,500 support Candidate A?

(c) Do the ratios from parts (a) and (b) represent a proportional relationship?

DEVELOPING SKILLS AND UNDERSTANDING

(3) The tables below each give the dimensions of four different rectangles. Write in the blank if the four rectangles in each set are “Proportional” or “Not Proportional”.

(a) ____________

	Width	Length
Rectangle 1	17	110.5
Rectangle 2	23.4	152.1
Rectangle 3	33	214.5
Rectangle 4	52.2	339.3

(b) ____________

	Width	Length
Rectangle 1	7.4	15.5
Rectangle 2	17	40.8
Rectangle 3	23.4	68.5
Rectangle 4	33	72.6

(4) A marine biologist would like to feed some dolphins a mix of fish that consists of 9 parts cod to 4 parts mackerel. List a combination that would be an acceptable mixture of these fish.

(5) Identify the proportions that have the same solution as the one below. There may be more than one correct answer.

\[\dfrac{3}{x} = \dfrac{24}{56}\nonumber \]

(i) \(\dfrac{x}{3} = \dfrac{24}{56}\)

(ii) \(\dfrac{3}{24} = \dfrac{x}{56}\)

(iii) \(\dfrac{24}{3} = \dfrac{56}{x}\)

(iv) \(\dfrac{x}{24} = \dfrac{3}{56}\)

(v) \(\dfrac{x}{3} = \dfrac{56}{24}\)

(6) Solve the following proportions (round to one decimal place):

(a) \(\dfrac{x}{5} = \dfrac{10}{13}\)

(b) \(\dfrac{3}{4} = \dfrac{15}{2x}\)

(7) Erica would like to bake an 8-pound roast for a family gathering. The cookbook tells her to bake a 5-pound roast for 135 minutes. Create and solve a proportion that would allow Erica to cook her 8-pound roast.

(8) Cefaclor is a medication used for infections. It is often given in liquid form by mixing the powdered medication with a fluid. A pharmacist is mixing a dosage for a child. The instructions indicate that 125 mg of the medication should be mixed with 5 mL of fluid. If the child only requires a dosage of 100 mg of Cefaclor, how much fluid should the pharmacist use?

MAKING CONNECTIONS ACROSS THE COURSE

(9) A company is making pennants or flags for a sports team. The team wants small versions for fans and large versions that will fly over the stadium. The dimensions of the small version are shown below.

(a) The large version needs to be 12.5 feet across the base (the short side of the triangle). How long should it be?

(b) How much material will be used for the large version of the flag? Round up to the nearest tenth of a square foot to ensure the company has enough material.

(10) A staircase is made up of individual steps that should be consistent in height and width. The height of each step is called the rise, and the width of the step is called the run.

(a) The staircase below is made up of four steps each with a rise of 6.5" and a run of 8.25". Find the (i) height and (ii) depth of the entire staircase.

(b) Builders have to follow guidelines on the rise and run of stairs when building a staircase. One acceptable ratio is a rise of 7-3/4 inches for a run of 9-3/4 inches. If a builder is using this ratio to build a staircase that is 15 feet high, how deep will the staircase need to be (d in the drawing below)? Note that the drawing does not show the correct number of steps. Round to the nearest tenth of a foot.

Graphing on a Coordinate Plane

Now, you will once again return to graphing on a coordinate plane. You may have noticed that the two axes split the coordinate plane into four sections. These are called quadrants and are numbered using Roman numerals as shown below.

For practical reasons, only a small part of the coordinate plane can be shown, but understand that the axes can go on infinitely in all four directions. The scale of the grid tells you which numbers are included in the portion of the plane that is shown. You can change the scale to make graphs with very large or very small numbers. The scale on a single axis must be consistent. In other words, if the distance between the gridlines represents five units on one part of the horizontal axis, then that same distance must always represent five units on that axis. However, the vertical and horizontal axes can have different scales as in the example below. As you have seen with other types of graphs, it is important to pay close attention to the scale.

(11) Place the following points on the graph. Label each with its ordered pair.

(a) (–14, 7)

(b) (0, 9)

(c) (13, –6)

(12) Indicate if each statement is true or false.

(a) The point (–7, –5) is in Quadrant II.

i) True ii) False

(b) The point (0, 5) lies on the vertical axis.

i) True ii) False

(c) All the points in Quadrant IV have a positive horizontal coordinate and a negative vertical coordinate.

i) True ii) False

(d) The points (20, 12) and (20, 200) lie on the same horizontal line.

i) True ii) False

Many applications use only positive numbers. In these cases, only Quadrant I of the graph is usually shown because that is the only quadrant that is used. An example of this is given below.

(13) You learned about the golden ratio in Exercise 3.5. A rectangle whose dimensions match the golden ratio is called a golden rectangle. The graph below shows the widths and lengths of golden rectangles.

(a) Based on the graph, is a rectangle with a width of 17 inches and a length of 30 inches a golden rectangle?

i) Yes ii) No

(b) Use the graph to complete the table of values below. Estimate to the nearest whole number.

Width	Length
5	(i)
8	(ii)
(iii)	20
(iv)	31