# Mix problems

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Problem #1

The starting salaries for teachers in Broward County for selected years are given in the table. Use this

information to answer the questions below. Show work for all calculations and give clear concise

explanations to all questions.

School Year Starting Salary

1995

2000

2005

*2014

*Projected

$27,910

$33,427

$37,000

$41,400

a) Find and interpret the average rate of change of starting salaries for teachers in Broward County

between 1995 and 2014. Please write as a decimal rounded to the nearest penny.

b) Assume starting salaries in the future increase at the same rate, use the years 1995 and 2014 to find a

linear model (equation) that can be used to predict future starting salaries (let x = 0 in year 1995). Use

your answer from part a for the slope and use function notation with s(x) representing starting salary.

c) Use your model from part b to predict starting salaries for teachers in Broward County for the year

2025 (let x = 0 in year 1995). Please write as a decimal rounded to the nearest penny.

d) Use your model from part b and algebra to predict the year in which starting salaries for teachers in

Broward County will reach $50,000 (let x = 0 in year 1995).

Problem #2

Justin started a lawn maintenance business. His basic rate to maintain an average size lawn is $30. It cost

Justin $1,250 to start the business (trailer, equipment, etc.) and he estimates that his cost per lawn is $1.25

(gas, oil, blades, etc.). Use this information to answer the questions below.

a) Write the cost function in terms of c(x) and the revenue function in terms of r(x) for Justin's business.

b) Set up an equation and use algebra to find the number of lawns that Justin has to maintain for his

business to break-even (cost equal to revenue)?

c) The profit function p(x) can be found by subtracting the cost function from the revenue function

p(x) = r(x) - c(x). Write the profit function in simplest form for Justin's business in terms of p(x).

d) Use the profit function from part c to evaluate and interpret p(500).

e) Set up an equation and use algebra to find when Justin's business will make a profit of $25,000?

f) Sketch the cost and revenue functions on the same graph.

Problem #3

Use the shortcut as demonstrated in class to perform the following conversion between number bases.

Please show all work and use the proper notation in your final answer.

A4B3C2D1sixteen to Binary

Problem #4

First choose a five digit number without repeating digits and a three digit number without repeating digits

(ex. 15,294 and 205). Then in the space below, show how the lattice method would be used to multiply

these numbers. Please show all work as if you were teaching someone how to use this method for the

first time (clearly identify the answer). Also, briefly explain why this method was discussed in this class.

Problem #5

First choose a two digit number greater than 32 and not equal to a power of 2 and a three digit number

greater than 100 and not equal to a power of two (ex. 43 and 105). Then in the space below, demonstrate

how the ancient Egyptian's would multiply these numbers. Please show all work as if you were teaching

someone how to use this method for the first time (clearly identify the answer).

Problem #6

First choose and write a three digit Babylonian number (use Babylonian numeral) without repeated digits.

Then in the space below, show step by step how to convert the Babylonian number to Hindu-Arabic.

Then convert the Hindu-Arabic to Mayan numerals (make sure to use appropriate spacing for the

Babylonian and Mayan numeral as demonstrated in class).

Problem #7

A local sandwich store has a fixed weekly cost of $525 and a variable cost of $.55 per sandwich. Use this

information to answer the questions below.

a) Write the cost function in terms of c(x) for this sandwich store.

b) If the weekly revenue function is given by the function ( )

, write the stores

weekly profit function in simplest form and in terms of p(x). hint: p(x) = r(x) - c(x)

c) Use the profit function p(x) from part b to find the number of sandwiches the store should make and

sell each week to maximize profits. Also, what is the maximum weekly profit in dollars to the nearest

penny?

d) Find and interpret p(1000), p(1200), p(1300), and p(1500). Why is it not better for this sandwich

shop to sell more sandwiches?

Problem #8

The risk of having a car accident increases exponentially as the concentration of alcohol in the blood

increases. The risk can be modeled by

where x is the blood alcohol concentration and R,

given as a percentage, is the risk of having a car accident.

a) In the state of Florida, the legal blood alcohol level (BAL) is 0.08. At this level, what is the risk of a

car accident? Give the answer rounded to two decimal places.

b) At what blood alcohol level will the risk of having a car accident increase to 25%? Give the answer

rounded to three decimal places.

Your blood alcohol level can be approximated by the formula (

) where D =

number of drinks consumed with alcohol content equivalent to one 12 oz beer, W is the weight of the

person, H = number of hours since last drink, and r = .79 for men and r = .71 for women.

c) Based on your gender and your weight, would you over the legal limit immediately after having 4

drinks (H=0)? Give the answer rounded to three decimal places. If you are over the limit, how many

hours would you have to wait before you can legally drive?

d) Based on your BAL immediately after 4 drinks (H=0), if you did drive, what is the risk of you having

a car accident? Give the answer rounded to two decimal places.

Problem #9

Xanax is a tranquilizer used in the short-term relief of symptoms of anxiety. Its half-life in the

bloodstream is 12 hours.

a) Find an exponential decay function that models the decay of Xanax in the bloodstream. Round the

decay constant k to six decimal places.

b) Use the model from part a to find what percentage of the original dose of Xanax is present in the

bloodstream after 18 hours. Round the percent to the nearest hundredth.

c) Use the model from part a to find out how long it will take for Xanax to decay to 15% of its original

dosage. Round to the nearest hour.

Problem #10

A ball is propelled upward from ground level. After t seconds, its height in feet is defined by the equation:

h(t) = height in feet and t = time in seconds

a) What is the maximum height that the ball will reach and how long will it take to reach maximum

height (please include the appropriate units of measure)?

b) Find h(5). Interpret these results. Is the ball on the way up or on the way down at this time?

c) Find h(t) = 112. Interpret these results. Is the ball on the way up or on the way down at this time?

d) How many seconds will it take for the ball to hit the ground? Use algebra to justify your answer.

#### Solution Preview

1)

Year (x) Starting Salary(y)

1995 (0) $27,910

2000 (5) $33,427

2005 (10) $37,000

2014 (19) $41,400

a)

mean:

year:

X = (0+5+10+19)/4 = 8.5

Starting salaray:

Y = (27910+33427+37000+41400)/4 = $34,934.25

Linear relation:

y = s(x) = Bo + B1.x

Rate of change,

B1 = sum((xi-X)(yi-Y))/sum((xi-X)^2)

=> B1 = sum((0-8.5)*(27910-34934.25) + (5-8.5)*(33427-34934.25) + (10-8.5)*(37000-34934.25) + (19-8.5)*(41400-34934.25))/sum((0-8.5)^2+(5-8.5)^2+(10-8.5)^2+(19-8.5)^2)

=> B1 = $690.21

Hence, rate of change of starting salary = $690.21/year --Answer

b)

s(x) = Bo + B1.x

Bo = Y - B1.X = 34934.25 - 690.21*8.5 = 29067.50

As 1995 == 0

Hence, starting salary (y) as function of year (x) equation:

s(x) = 29067.50 + 690.21(year - 1995) --Answer

c)

year = 2025, starting salary

s(x) = 29067.50 + 690.21(2025 - 1995) = $49,773.80 --Answer

d)

satrting salary = $50,000

s(x) = 50000 = 29067.50 + 690.21(year - 1995)

=> year = 1995 + (50000-29067.50)/690.21 = 2025.33 --Answer

2)

Fixed cost, fc = $1,250

Variable cost, vc = $1.25/lawn

Basic rate, br = $30/lawn

a)

Cost funtion in terms of number of lawns, x:

c(x) = 1250 + 1.25x --Answer

Revenue function,

r(x) = 30x

b)

For break-even point,

c(x) = r(x)

1.25x + 1250 = 30x

=> 28.75x = 1250

=> x = 1250/28.75 = 43.48 --Answer

c)

Profit function,

p(x) = r(x) - c(x)

=> p(x) = 30x - (1250 + 1.25x)

=> p(x) = 28.75x - 1250

d)

p(500) = 28.75*500 - 1250 = ...

#### Solution Summary

A few problems of finance and number systems, from linear algebra are solved.