3.4.2: Exercise 3.4

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MAKING CONNECTIONS TO THE COLLABORATION

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

(i) When using variables, it is important to know what they represent and what units should be used with them.

(ii) When using variables, it is only important to know what numbers to substitute in for them.

(iii) A subscript is a label on a variable.

(iv) Braking distance is affected by many factors.

DEVELOPING SKILLS AND UNDERSTANDING

Mathematical Terminology

Formulas are a type of algebraic equation. You have probably seen algebraic equations such as “y = x + 3” in previous math classes. Each side of the equation is called an algebraic expression. So “x + 3” is an expression and “y” is an expression. An equation is a statement that two expressions are equal. The purpose of such an equation is to define a sequence of calculations using a shortcut language. In this example, the equation “y = x + 3” means:

(1) Start with x.

(2) Add three to x.

(3) The result is y.

The word formula is usually used to express important and non-changing relationships, especially in contexts such as science, business, medicine, sports, or statistics. For example the area of a rectangle, A = L ⋅ W, is a formula because the relationship between area and the length and width of a rectangle is always the same. It is also an equation, but that wording is less common.

Suppose you had a situation in which you make $12 per hour. This relationship could be written algebraically as P = 12h where P is your pay in dollars, and h is the number of hours you work. This would be called an equation instead of a formula because if you got a raise, the relationship would change. You also might call the equation a model because it models a situation mathematically.

In other math classes, you might see problems like the one shown below. Each line represents a simplification of the line above.

Example: Evaluate the expression 3x²+ 2y if x = 3 and y = 5.

3(3)² +2(5)

3(9) + 10

27 + 10

These types of problems are not included in this course because the focus is on using mathematics in a more every-day context. However, you should recognize that this type of expression uses the same skills that you used when working with formulas. Units are not involved, so the first step is to recognize that the values can replace the variables and the order of operations is applied to simplify the form of the problem.

(2) In the collaboration, you investigated the relationship between velocity and braking distance. You will now investigate the relationship between the coefficient of friction and the braking distance.

Recall that the formula for the braking distance of a car is $d = \dfrac{V^{2}_{0}}{2g(f + G)}$

(a) Define each variable:

(i) V₀

(ii) d

(iii) G

(iv) f

(v) g

(b) Which of the variables have no units?

(d) To investigate the relationship between the coefficient of friction and the braking distance, you need to hold the other variables fixed. Let G = 0.02. Which of the following is a correct interpretation of the value G = 0.02?

(i) The grade of a road is 0.02%, which is a vertical increase of 0.02 feet for every 1 foot of horizontal increase.

(ii) The grade of a road is 0.02%, which is a vertical increase of 0.02 feet for every 100 feet of horizontal increase.

(iii) The grade of a road is 2%, which is a vertical increase of 2 feet for every 1 foot of horizontal increase.

(iv) The grade of a road is 2%, which is a vertical increase of 2 feet for every 100 feet of horizontal increase.

(e) Let V₀ = 72 mph and use the value of G = 0.02. Which of the following expressions represents the simplified form of the formula using these values? (Hint: Convert V₀ and g into ft/s.)

(i) $d = \dfrac{11,151.36}{(64.4f + 0.02)}ft$

(ii) $d = \dfrac{11,151.36}{64.4f} + 8,657.89ft$

(iii) $d = \dfrac{11,151.36}{(64.4f + 1.288)}ft$

(iv) $d = \dfrac{0.02}{(64.4f + 1.288)}ft$

(3) (a) Use the formula you found in Question 2(e). Complete the table of values for f and d (in feet). Use the values of f given in the table. Perform one of the calculations on paper showing the units. You may then use technology to complete the table. Round solutions to 3 decimal places.

f	d (feet)
0.30	(i)
0.50	(ii)
0.70	(ii)
0.90	(iv)

(b) The four values of f correspond to the coefficient of friction for four road and tire conditions: an icy road with fair tires, a very good road with great tires, an asphalt road with fair tires, and a wet road with fair tires. Match the coefficients of friction to the appropriate conditions by looking at the braking distance required.

(i) Icy road with fair tires, f =

(ii) Very good road with great tires, f =

(iii) Medium quality road with fair tires, f =

(iv) Wet road with fair tires, f =

(c) The coefficient of friction (f) is increasing at a constant rate, since each value is 0.2 more than the previous value. How is d changing as f increases at a constant rate?

(i) The stopping distance is decreasing.

(ii) The stopping distance is constant.

(iii) The stopping distance is increasing.

(4) Lenders such as banks, credit unions, and mortgage companies make loans. The person receiving the loan usually pays the loan off in small payments over a long period of time. The lender earns money by charging interest, which is based on a percentage of the amount that is borrowed. There are different types of interest. Car loans are usually calculated using the formula for simple interest. The total amount repaid is based on the interest and the value of the original loan, called the principal. The formula for the total dollars needed to repay the loan, with interest, is found using the formula

\[A = P + P\cdot r \cdot t \nonumber \]

where

A is the amount (total principal plus interest) required to repay the loan.
P is the amount borrowed, the principal.
r is the annual interest rate, quoted as a percent, but used as a decimal in the formula.
t is the time, in years, taken to repay the loan (six months would be 1/2 year).

Suppose you get a loan of $10,000 at an annual interest rate of 6.25%.

(a) Use the given information to write the formula for the total amount to be repaid in t years.

(b) Complete the table of values below that shows the payoff amount after certain amounts of time.

	t (years)	A ($)
0	(i)	(vi)
4 months	$\dfrac{1}{3}$	(vii)
6 months	(ii)	(viii)
1 year	(iii)	(ix)
3 years	(iv)	(x)
6 years	(v)	(xi)

MAKING CONNECTIONS ACROSS THE COURSE

(5) The tuition at a daycare center is based on family income. A reduced tuition is offered to families with lower incomes, which is a form of financial aid called a subsidy. There are three levels of tuition:

Full subsidy—the family does not pay any tuition
Partial subsidy—the family pays part of the tuition
No subsidy—the family pays the full tuition

The data for the daycare center for each age level is given below. Answer the questions below. Round to the nearest whole number.

	Full Subsidy	Partial Subsidy	No Subsidy	Total
3-year-olds	17	13	8	(i)
4-year-olds	22	14	15	(ii)
5-year-olds	15	16	11	(iii)
Total	(iv)	(v)	(vi)	(vii)

(a) Complete the last column and last row of the table of the table above.

(b) What percentage of 3-year-olds received a full or partial subsidy? Round to the nearest whole percent.

(d) What percentage of the children are 3 years old?

(e) The daycare center receives federal funding based on the number of children receiving a subsidy. The amount of federal funding for the current term can be calculated using the formula below. Calculate the center’s federal funding for the current term.

Funding = 1,530F + 1,750P + 1,875N

where:

F = number of children receiving a full subsidy

P = number of children receiving a partial subsidy

N = number of children receiving no subsidy

(6) In Collaboration 1.7, you used a formula that was written as steps in a form to calculate self-employment taxes for different people. Formulas are often written in this way. One example is the Expected Family Contribution (EFC) Formula, which is used to determine if a college student is eligible for financial aid. The EFC has many different sections that each use different calculations. One section of the 2022–23 form is shown below.¹⁴ This section includes lines 45–50.

Student’s Contribution from Assets

45. Cash, savings, and checking
46. Net worth of investments If negative, enter zero	+
47. Net worth of business and/or investment farm If negative, enter zero	+
48. Net worth (sum of lines 45 through 47)	=
49. Assessment rate	×	0.20
50. STUDENT’S CONTRIBUTION FROM ASSETS	=

(a) Calculate the “Student’s Contribution from Assets” given the following information.

Cash: $500 Investments: loss of $2,000

Savings: $3,240 Business: $0

Checking: $732

(b) Assuming that the “Net Worth of Investment” (N_I) and the “Net Worth of Business or Farm” (N_B) are positive, write a formula that summarizes the calculation in this form using the following variables:

C = Cash including savings and checking

N_I = Net worth of investment

N_B = Net worth of business or farm

S = Student’s contribution from assets

_____________________________________

¹⁴ https://fsapartners.ed.gov/sites/default/files/2021-08/2223EFCFormulaGuide.pdf

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