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New Westminister City Council is adding 24 streetlights each year for five years. Currently, there are 468 streetlights. Define your variables and write both a recursive and explicit formula for the total number of streetlights over the next five years.
New Westminister City Council is adding 24 streetlights each year for five years. Currently, there are 468 streetlights. Define your variables and write both a recursive and explicit formula for the total number of streetlights over the next five years.
Let $S_n$ be the number of streetlights in year $n$, with $S_0 = 468$. Recursive formula: $S_{n+1} = S_n + 24$. Explicit formula: $S_n = 468 + 24n$.
New Westminister City is budgeting for new streetlights costing $4,000 each, increasing 6% annually. With storage for only 25 streetlights, what is the total cost for installing two streetlights per month for five years? State any assumptions.
New Westminister City is budgeting for new streetlights costing $4,000 each, increasing 6% annually. With storage for only 25 streetlights, what is the total cost for installing two streetlights per month for five years? State any assumptions.
Assuming immediate installation upon arrival to avoid storage costs: 2 lights/month * 5 years * 12 months/year = 120 lights. Considering the 6% annual price increase, the cost escalates over time. To calculate the exact total cost, we would sum the cost of each light at the price of the year it was installed in:
Total Cost = $\sum_{i=1}^{120} 4000(1.06)^{\lceil\frac{i}{24}\rceil - 1}$
Which investment yields better returns: 8.1% APR compounded annually, or 7.9% compounded monthly? Justify your answer.
Which investment yields better returns: 8.1% APR compounded annually, or 7.9% compounded monthly? Justify your answer.
We need to calculate the effective annual rate (EAR) for the 7.9% compounded monthly. $EAR = (1 + \frac{0.079}{12})^{12} - 1 \approx 0.0819$ or 8.19%. Thus, 7.9% compounded monthly is better.
Friends introduce 50 koi to a pond that can support 1000 koi, with a natural growth rate of 40% per year. Use a recursive logistic growth model to determine the koi population each year for ten years.
Friends introduce 50 koi to a pond that can support 1000 koi, with a natural growth rate of 40% per year. Use a recursive logistic growth model to determine the koi population each year for ten years.
Langley Township's population is 137,400 and grows at 2.4% annually. When will the population reach 200,000?
Langley Township's population is 137,400 and grows at 2.4% annually. When will the population reach 200,000?
You buy a $35,000 car with an 8% APR loan over 7 years. After four years, how much do you still owe, and what is the total interest paid? If the car devalues by 20% yearly, what is it worth after four years?
You buy a $35,000 car with an 8% APR loan over 7 years. After four years, how much do you still owe, and what is the total interest paid? If the car devalues by 20% yearly, what is it worth after four years?
A bacteria culture starts with 300 bacteria, growing to 500 in 4 hours. Assuming exponential growth: a. Write a recursive formula for the number of bacteria. b. Write an explicit formula. c. How many bacteria after 1 day? d. How long to triple the initial size?
A bacteria culture starts with 300 bacteria, growing to 500 in 4 hours. Assuming exponential growth: a. Write a recursive formula for the number of bacteria. b. Write an explicit formula. c. How many bacteria after 1 day? d. How long to triple the initial size?
You have 500 M&M's and eat 1/4 of the remaining candies each day. Is the number of candies decreasing linearly or exponentially? Write an equation to model candies left after n days.
You have 500 M&M's and eat 1/4 of the remaining candies each day. Is the number of candies decreasing linearly or exponentially? Write an equation to model candies left after n days.
One hundred trout are seeded into a lake with a 70% annual growth rate, constrained by a maximum of 2000. Using the logistic growth model: a. Write a recursive formula. b. Calculate trout count after 1 and 2 years.
One hundred trout are seeded into a lake with a 70% annual growth rate, constrained by a maximum of 2000. Using the logistic growth model: a. Write a recursive formula. b. Calculate trout count after 1 and 2 years.
Inflation decreases money's value by 5% yearly. If you have $1 this year, it buys $0.95 worth of goods next year. How much will $100 buy in 20 years?
Inflation decreases money's value by 5% yearly. If you have $1 this year, it buys $0.95 worth of goods next year. How much will $100 buy in 20 years?
Flashcards
Recursive Formula
Recursive Formula
A formula where each term is defined using preceding terms.
Explicit Formula
Explicit Formula
A formula where any term can be calculated directly without knowing the preceding terms.
Logistic Growth Model
Logistic Growth Model
Models population growth considering carrying capacity (K).
APR (Annual Percentage Rate)
APR (Annual Percentage Rate)
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Depreciation
Depreciation
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Compound Interest
Compound Interest
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Exponential Growth
Exponential Growth
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Linear Growth
Linear Growth
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Carrying Capacity (K)
Carrying Capacity (K)
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T-Bill
T-Bill
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Study Notes
- New Westminster City Council plans to install 24 streetlights annually for the next five years, with 468 streetlights currently.
- A recursive and explicit formula is needed to calculate the number of streetlights over the next five years.
City Budgeting for Streetlights
- The current cost for an installed streetlight is $4,000, increasing by 6% annually.
- The city yard can store only 25 streetlights.
- It is important to calculate the total cost for installing streetlights at a rate of 2 per month for 5 years, considering storage limitations and cost increases using defined assumptions.
Investment Options
- It is important to Determine whether an investment with 8.1% APR compounded annually is better than one with 7.9% compounded monthly and provide a justification.
Koi Population Growth
- A certain pond can support about 1000 koi and starts with 50 koi, with a natural growth rate of 40% per year.
- It is important to use recursive logistic growth to determine the koi population each year for the next 10 years.
Township population
- Langley Township has a population of 137,400, growing at 2.4%. Determine how long it will take for the population to reach 200,000.
Car Loan and Depreciation
- A car is purchased for $35,000 with a 7-year loan at 8% APR.
- Calculate the remaining balance after four years, the amount paid in interest, and the car's value after four years with a 20% annual devaluation.
Bacteria Growth
- A bacteria culture starts with 300 bacteria and grows to 500 after 4 hours.
- Develop a recursive formula for the bacteria number.
- Develop an explicit formula for the bacteria number.
- Estimate how many bacteria there will be in 1 day; determine how long it takes for the culture to triple in size, assuming exponential growth.
M&M Consumption
- Someone starts with 500 M&M candies and eats ¼ of the remaining candies each day.
- Find whether the number of candies left changes linearly or exponentially.
- Create an equation to model the number of candies left after n days.
Trout Population
- A lake is seeded with 100 trout, with a 70% annual growth rate and a maximum sustainable population of 2000.
- Develop a recursive formula for the number of trout.
- Calculate the number of trout after 1 and 2 years using the logistic growth model.
Inflation Impact
- Inflation decreases the value of money by 5% each year.
- Find how much $100 will buy in 20 years, given the annual 5% decrease in value.
Fish Harvesting
- 2000 fish currently reside in a lake and the fish population grows by 10% each year, but 100 fish are harvested annually.
- Develop a recursive equation for the number of fish in the lake after n years.
- Calculate the population after 1 and 2 years and find whether the population appears to be increasing or decreasing.
- It is necessary to determine the maximum number of fish that could be harvested each year without causing the fish population to decrease in the long run.
One-Time Interest Loan
- A friend lends $200 for a week with a 5% one-time interest.
- It is important to calculate the total amount to repay.
T-Bill Investment
- Consider buying a $5000 face value 26-week T-bill; determine the most you should pay to earn at least 1% annual interest.
Compound Interest
- Find the future value in 20 years of $1000 deposited in an account earning 7% interest compounded annually.
- Determine how much to deposit now to have $6,000 in 8 years, assuming a 6% interest compounded monthly.
- Calculate the future savings, total contributions, and total interest earned on monthly deposits of $200 into an account earning 3% interest compounded monthly over 30 years.
- Determine the required monthly deposit, total contributions, and total interest earned to have $800,000 for retirement in 30 years with a 6% interest account.
Retirement Withdrawals
- It is important to determine the initial amount needed in an account earning 8% interest to withdraw $30,000 each year for 25 years.
- Also find the total amount withdrawn and the total interest earned.
- Calculate the monthly withdrawals possible for 20 years from a $500,000 retirement account earning 6% interest.
Mortgage Affordability
- One can afford a $700 per month mortgage payment on a 30-year loan at 5% interest.
- Determine how big of a loan can be afforded, the total amount paid, and the total interest paid.
- Calculate the loan amount, monthly payments at 5% interest, and monthly payments at 6% interest for a $200,000 home purchase with a 10% down payment and a 30-year loan.
- With a starting deposit of $6,000 in an account earning 4% compounded monthly, calculate how long it will take for the account to grow to $10,000.
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