# Mathematical Induction Questions and Answers

Math

Mathematical InductionFlu is spreading exponentially at a school. The number of new flu patients can be modeled using the equation F = 10e0.12d, where d represents the number of days since 10 students had the flu. (a) How many days will it take for the number of new flu patients to equal 50? Determine your answer algebraically using the natural logarithm. Round your answer to the nearest day.

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Mathematical InductionCarly wants to invest $1000 in an account that earns 8% interest, compounded quarterly. How many years will it take for he account to have a value of $5000? Round to the nearest whole number.

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Mathematical InductionFind S, through S, and then use the pattern to make a conjecture about S, Prove the conjectured formula for S, by mathematical induction.
Sn:1/22+1/66+1/132+1/220+...+1/11n(n+1)
Find S, through S5-
(Type an integer or a simplified fraction.)
S₁ =

Math

Mathematical InductionDetermine the a) down payment, b) amount of the loan, c) monthly payment, and d) finance charge.
Jason Wagner obtains an installment loan to buy a $12,000 Honda Accord. His down payment is 25%, and the APR is 9% for 36 months.

Math

Mathematical InductionA 75 L gas tank has a leak. After t hours, the remaining volume, V, in litres is V(t) = 75 (1 - t/24) ²,0≤ t ≤ 24. Use the product rule to determine how quickly the gas is leaking from the tank when the tank is 60% full of gas.

Math

Mathematical Induction-6z² +6z +140/ (z + 5)² (3z-5)
Determine the correct partial fraction decomposition for the expression above.
Provide your answer below:

Math

Mathematical InductionYou buy fishing equipment costing $268. State taxes are 6.5% and the federal excise tax on this item is 11.45%. What is the total purchase price? (Round your answer to the nearest cent.)
$285.42
$296.87
$298.69
$316.11

Math

Mathematical InductionUse mathematical induction to prove the statement for all positive integers n.
2+14+26+...+(12n-10) = n(6n-4)
What is the first step in a mathematical induction proof?
A. Show that S₁ is true.
B. Show that So is true.
C. Show that Sk is true.
D. Show that Sk + 1 is true.
Show that 2+14+26++ (12n-10) = n(6n-4) for n = 1. Write the statement S₁.
2= (Do not simplify.)
Simplify S₁ on the right.
2=
Is S₁ a true statement?

Math

Mathematical InductionA small metal bar, whose initial temperature was 10° C, is dropped into a large container of boiling water. How long will it take the bar to reach 70° C if it is known that its temperature increases 2º during the first second? (The boiling temperature for water is 100° C. Round your answer to one decimal place.)
How long will it take the bar to reach 96° C? (Round your answer to one decimal place.)

Math

Mathematical InductionSuppose a certain company gave raises of 3% each year for three years followed by raises of 5% for the next two years.
a. If an employee has an annual salary of $52,500 after those 5 raises, what was their starting salary? Show a formula or some calculations and explain in words what you are doing. The anual Salary after 5 years = x dollars.1.051.030.
b. What was their 'average annual growth' over those 5 years? In other words, if they had the same percentage raise each year, but had the same increase over the 5-year span, what would that constant 1 percentage raise be?

Math

Mathematical InductionA population numbers 11,000 organisms initially and decreases by 2.9% each year.
Suppose P represents population, and t the number of years of growth. Write an exponential model, P, in the form P = a b^t

Math

Mathematical InductionSuppose you pay $2.00 to roll a fair die with the understanding that you will get back $4.00 for rolling a 2 or a 1, nothing otherwise.
What is your expected net winnings?
$4.00
-$2.00
-$0.67
$2.00

Math

Mathematical InductionCompute the area of the region R bounded by the curve
r = 1- cosθ. This curve was sketched in the previous homework (and
you can refer to that sketch). Use the double integral method (polar
integrals; section 14.3), that is, integrate function 1 over the region.

Math

Mathematical InductionDan is putting money into a savings account. He starts with $350 in the savings account, and each week he adds $30.
Let S represent the total amount of money in the savings account (in dollars), and let W represent the number of weeks Dan has been adding money. Write an
equation relating S to W. Then use this equation to find the total amount of money in the savings account after 17 weeks.

Math

Mathematical InductionSuppose that the profit obtained from the sale of x fish
-fry dinners
is given by P(x) = -0.025x² + 8x - 40. What is the
marginal profit, in
dollars from the sale of the 101st fish-fry dinner? Write
your answer
to two decimal places and donot use the dollar sign symbol.

Math

Mathematical InductionFind the monthly payment (in $) and the total interest (in $) for a mortgage of $48,000 at 5 3/4% for 30 years. Use this table and give the total interest as your answer. (Round your

Math

Mathematical InductionThe area of a square is increasing at a rate of 12 centimeters squared per second. Find the rate of change of the side of the square when it is 2 centimeters.

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Mathematical InductionAn astronomer hosts a stargazing party for $775. If the group tips the astronomer 15%, what is the total amount that the astronomer earns for the stargazing party? Enter the correct answer in the box.

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Mathematical InductionHow long, to the nearest year, will it take an investment to double if it is continuously compounded at 9% per year?

Math

Mathematical Induction6. In 1848 the mathematician Alphonse de Polignac conjectured that "Every odd number can be expressed as the sum of a power of 2 and a prime."
(a) Verify the conjecture for the numbers 61 and 119.
(b) Show that the number 509 demonstrates this conjecture is false.

Math

Mathematical InductionAn amount of $23,000 is borrowed for 11 years at 8% interest, compounded annually. If the loan is paid in full at the end of that period, how much must be paid back? Use the calculator provided and round your answer to the nearest dollar.

Math

Mathematical InductionFor f(x) = x + 2 and g(x) = 2x + 3, find the following functions.
a. (fog)(x); b. (gof)(x); c. (fog)(-2); d. (gof)(-2)

Math

Mathematical InductionUPS charges $7 for the first pound and $0.20 for each additional pound. FedEx charges $5 for the first pound and $0.30 for each additional pound. How many additional pounds, p, will it take for UPS and FedEx to cost the same? What is the cost? Write and solve an equation to show your work.

Math

Mathematical InductionA particle is moving along the x-axis. The position of the particle at time t is given by
x(t) = t³ - 6t² + 9t - 2, 0 ≤ t ≤ 5.
Find the total distance the particle travels in 5 units of time.

Math

Mathematical InductionIt is known that the number of trees in a small forest will decrease by 5% each year unless some new
trees are planted. At the end of each year, 600 new trees are planted to the forest. At the start of
2021, there are 8,200 trees in the forest.
a) Show that there will be roughly 9060 trees in the forest at the start of 2026.
b) Find the approximate number of trees in the forest at the start of 2041.

Math

Mathematical InductionThe cost of parking at the Center City Parking Garage is $5.50 for the first hour or part of
an hour and $2.75 for each additional half hour or part of a half hour. The maximum cost
for 24 hours is $50. Does the cost of parking vary directly as the number of hours? Explain
your answer.

Math

Mathematical InductionUse polynomial long division to rewrite the following fraction in the form g(x) +r(x)/d(x),where d(x) is the denominator of the original fraction, q) is the quotient, and r(x) is the remainder.
Correct
3x - 15x²³ +21x²+2x+3
3x² +3

Math

Mathematical InductionGiven the function f(t) = (t - 6) (t + 3)(t - 5)
its f-intercept is
its t-intercepts are

Math

Mathematical InductionTim is trying to decide which loan is better. Bank A offers
compounded weekly, while Bank B offers a 10 year loan at a rate of 5.75% compounded continuously.
15 year loan at a rate of 4.5%
Tim wants the loan that will result in the lowest total owed. If he is borrowing $4,800, which loan
should he take?

Math

Mathematical InductionThere were 16 necklaces that each had 31 beads. If Shana took all the beads off and put an equal amount on 7 chains, how many beads would be left over?
A. O
B. 2
C. 4
D. 6

Math

Mathematical InductionTrain A has a speed 35 miles per hour greater than that of train B. If train A travels 300 miles in the same times train B travels 195 miles, what are the speeds of the two trains?

Math

Mathematical InductionBen Franklin's gift of $4,000 to New York City grew to $5,000,000 in 200 years. At what interest rate compounded annually would this growth occur?
The interest rate would be %.
(Round to the nearest hundredth.)

Math

Mathematical InductionWilma can mow a lawn in 30 minutes. Bullwinkle can mow the same lawn in 90 minutes. How long
does it take for both Wilma and Bullwinkle to mow the lawn if they are working together?
minutes
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Math

Mathematical InductionIf Roy does a job in 10 hours and with the help of Kathryn they can do it together in 3 hours, how
long would it take Kathryn to do it alone?
Note: Write fractional answers as a fraction in lowest terms.

Math

Mathematical InductionWrite the negation of the following conditional statement:
"If your score is between 12 and 17, you will place into the first course."
If you do not place into the first course, then your score is not between 12 and 17
Your score is between 12 and 17, and you place into the first course
Your score is not between 12 and 17, and you do not place into the first course
Your score is between 12 and 17, and you do not place into the first course
Your score is not between 12 and 17, and you place into the first course
If your score is not between 12 and 17, then you do not place into the first course
Your score is not between 12 and 17, or you place into the first course
If you place into the first course, then your score is between 12 and 17

Math

Mathematical InductionEconomists speak of equilibrium, where the demand exactly equals the supply.
Your store supplies ear plugs with 1000 ear plugs per box (or per unit).
You know from historical data that your demand per unit can be written as d(x)=x+6
Likewise your supply can be written as s(x) = 3x-4
How many units would your store produce at equilibrium?

Math

Mathematical Induction(c) In another study, medical researchers concluded that more than 12% of elderly people have sleep apnea. Based on the confidence interval, does it appear that more than 12% of people aged 65 and over have sleep apnea? Explain.

Math

Mathematical InductionNoel has an initial account balance of $4,000. For the next five months, her account balance increases by $1,000 per month. After that, her account balance, b, decreases by $1,000 per month, t. Which statements are true for this situation?
The equation that models the account balance is b=-|11,000r-5,000| +9,000.
The equation that models the account balance is b=-|11,000t-9,000| +9,000.
The equation that models the account balance is b=-|11,000t-9,000| +4,000.
Noel's account balance will be $7,000 in 3 months and 7 months.
Noel's account balance will be $7,000 in 7 months and 11 months.

Math

Mathematical InductionThe number of bats in a colony is growing
After 3 years, there were 280 bats.
After 5 years, there were 1120 bats.
exponentially.
If the colony continues to grow at the same rate,
how many bats are expected to be in the colony
after 11 years? Do not include units in your
answer.

Math

Mathematical InductionIf the speed of a car with a turbo engine can be
modeled as a function of time using the parabolic
equation s(t) = 2t², how many seconds will it take
the car to exceed the 60 mph?
4
O5
06
07
8

Math

Mathematical InductionA quality control expert at LIFE batteries wants to test their new batteries. The design engineer claims they have a standard deviation of 86 minutes with a mean life of
505 minutes,
If the claim is true, in a sample of 120 batteries, what is the probability that the mean battery life would be greater than 523.8 minutes? Round your answer to four
decimal places.

Math

Mathematical InductionA manufacturing company currently uses 30 workers to load and unload trucks. Each worker can move an average of 14 boxes per minute. A robotics firm has built a robot that can load and unload boxes at the rate of 21 boxes per minute. How many robots will it take to do the same job as the 30 humans?
The required number of robots is
(Simplify your answer.)
***

Math

Mathematical InductionAn apartment complex currently houses 280 people who are spread out across buildings standing on 200,000 square feet of land. The owner of the complex is planning to expand by purchasing an additional square plot of land measuring 200 ft by 200 ft and constructing one more apartment building on this plot.
Assuming that the new building will house as many people as each of the other 4 buildings, which statement provides the best estimate of the effect this expansion will have in terms of the square feet of land per person living in the complex?
the square footage of land per person will decrease by about 140 ft²
the square footage of land per person will decrease by about 30 ft²
the square footage of land per person will not change
the square footage of land per person will increase by about 140 ft²

Math

Mathematical InductionAssume that in one country in 2005, 13 % of the population was 65 or older. By 2050, this percent is expected to be 21.55%. The percent of the population ages 25-34 in 2005 was 13.9%. That age group is expected to include 13.1395% of the population in 2050.
a. Assuming these population changes are linear, use the data for the 65 or over age group to write a linear equation. Then do the same for the 25-34 age group. Let x = 0 represent 2005 and x = 45 represent 2050. Express percents in decimal form.
For the population of 65 or older the equation is y =
For the population of 25-34 years the equation is y =

Math

Mathematical InductionAn undergraduate statistics class is assigned a project in which they are to sample 500 books from the university library and record the copyright year, the number of pages, and whether the book is hard cover or paperback. Determine the most appropriate way in which the students should illustrate the data on these variables.
Select one or more:
(a): Create a bar graph of number of pages.
(b): Create a pie chart of number of pages.
(c): Create a histogram of number of pages.
(d): Create a bar graph of hard cover vs. paperback
(e) : Create a histogram of hard cover vs. paperback

Math

Mathematical InductionYou must pick either the Block Method outline or the Point by Point method outline. If you use the block method outline, you will need 2-3 points per subject and in your third body paragraph you must tell me whad differences and similiarities you will be focusing on. For the point by point method, you will need to have 3 points and tell me how the two subjects connect.

Math

Mathematical InductionSuppose the extended Euclidean algorithm is applied to the numbers 240 and 46. It produces two integers a and b with the property 240a + 46b = GCD(240,46) What is the integer a?

Math

Mathematical InductionQ.72 Consider the following behaviours of function in (-1, 1)
Continuity IV.
I.
Increasing
II.
Decreasing
III.
Derivability
Which one of the following four functions exhibits atleast three of the four mentioned above?
(A) max (x, x³)
(B) max (x,x²})
(D) max (x, [x]}
(C) max (x,x|}
where [] denotes greatest integer function.

Math

Mathematical Induction8a. Toni and Sara make 50 cupcakes for a
bake sale.
Toni made 10 less cupcakes than Sara.
How many cupcakes did Toni make?
8b. Lucy and Effel make 80 chocolate
candies to sell.
Lucy made 12 less cupcakes than Effel..
How many cupcakes did Lucy make?