Solve for x:
Solve for x:
Find the integer interval in which the solution of the following equations will lie:
Consider the equation 3^{x} \times 3^{ n x} = 81.
Simplify 3^{x} \times 3^{ n x}.
Hence, solve the equation.
Solve 3^{x^{2} - 3 x} = 81.
Solve \dfrac{25^{y}}{5^{4 - y}} = \sqrt{125}.
Solve the following equation by using the substitution m = 4^{x}:
4^{ 2 x} - 65 \times 4^{x} + 64 = 0
Solve the following equation by using the substitution p = 2^{x}:
2^{ 2 x} - 12 \times 2^{x} + 32 = 0The graphs y = 2^{ 5 x} and y = 4^{x - 3} intersect at a certain point.
Find the x-coordinate of the point of intersection.
The astronomical unit (AU) is often used to measure distances within the solar system. One AU is equal to the average distance between Earth and the Sun, or 149\,597\,855 kilometres. The distance, d, of the nth planet from the Sun can be modelled by the formula:
d = \dfrac{3 \left(2^{n - 2}\right) + 4}{10}
where d is measured in astronomical units.
By substituting 4 for n, find the distance between Mars and the Sun in AU. Express your answer correct to two decimal places.
If the electricity bill is not paid by the due date, the company charges a fee for each day that it is overdue. The following table shows the fees:
| \text{Number of days after bill due } (x) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| \text{Overdue fee in dollars } (y) | 4 | 8 | 16 | 32 | 64 | 128 |
At what rate is the fee increasing each day?
If the bill is paid 3 days overdue, what overdue fee will it incur?
Find the equation that models the overdue fee, y, as a function of the number of days overdue, x.
A \$128 bill is overdue. How many days must it remain overdue for the overdue penalty to equal the cost of the bill itself?
Determine whether the each of the following functions is a model of exponential growth or exponential decay:
y = 2^{ 0.2 x}
y = 4^{ - 4.1 x }
y = 3 e^{ - 2 x }
y = \left(0.6\right)^{ 5 x}
The population, A, of aphids in a field of potato plants t weeks after intial observation is modelled by: A = 1000 \times 2^{t}.
State the initial aphid population.
Find the predicted aphid population after 5 weeks.
Find the predicted aphid population 2 weeks before initial observations.
The frequency f (Hz) of the nth key of an 88-key piano is given by f \left( n \right) = 440 \left(2^{\frac{1}{12}}\right)^{n - 49}.
Determine the frequency of the forty-ninth key.
Determine the frequency of the 40th key. Round your answer to the nearest whole number.
Find the value of n that corresponds to the key with a frequency of 1760 Hz.
The population, P, of a particular town after n years is modelled by P = P_0 \left(1.6\right)^{n}, where P_0 is the original population.
Find the population of the town after 3\dfrac{1}{2} years if its original population was 30\,000. Round your answer to the nearest whole number.
The mass in kilograms, M, of a baby orangutan at n months of age is given by the equation M = 1.8 \times 1.1^{n}, for ages up to n = 6 months.
What is the mass, M, of a baby orangutan at 3 months of age to one decimal place.
Complete the table. Round your answers to one decimal place.
| \text{months } (n) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| \text{mass } (M) |
Graph the function on a number plane.
Consider the table of values:
| \text{Number of days passed } (x) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Population of shrimp } (y) | 5 | 25 | 125 | 625 | 3125 |
Is the number of shrimp increasing by the same amount each day?
Find the equation linking population y and time x in the form y = a^{x}.
Sketch the graph of the equation.
The population, P, of a town in millions after t years is approximated by the formula P = 8 \left(1 + \dfrac{2}{100}\right)^{t}
Complete the table of values, round your answers correct to one decimal place:
| t | 5 | 10 | 15 | 20 | 25 | 30 |
|---|---|---|---|---|---|---|
| P |
Sketch the graph of the function P = 8 \left(1 + \dfrac{2}{100}\right)^{t}.
After how many years will the population reach 12.6 million?
In a laboratory, an antibiotic is tested on a sample of 8 bacteria in a petri dish. The number of bacteria is recorded, and the bacteria are found to double each hour.
Complete the table of values:
| \text{Number of hours passed }(x) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Number of bacteria }(y) | 8 | 32 |
Find the equation linking the number of bacteria (y) and the number of hours passed (x).
At this rate, how many bacteria will be present in the petri dish after 11 hours?
In a laboratory, a drug is tested on a sample of 2880 cancer cells. The number of cancer cells is recorded and the cancer cells are found to halve each week.
Complete the table of values:
| \text{Number of weeks passed }(x) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Number of cells }(y) | 2880 | 720 |
Find the equation linking the number of cancer cells (y) and the number of weeks passed (x).
According to this model, does the drug completely remove all the cancer cells? Explain your answer.
The following graph models the number of students, N, who have heard a rumour after t minutes:
The formula for this model is N = \dfrac{24}{1 + 24 \left(2^{ - t }\right)}
Use the graph to determine how many people in the class eventually heard the rumour.
When a heated substance such as water starts to cool, its temperature, T, after t minutes of being left to cool, is given by an exponential function.
Should temperature, T, be an increasing or decreasing function?
For positive integer values of a, the function T = b a^{ - t } is used to model the temperature, T, at time t. Substance P starts off at a temperature of 145 \degree C and is left to cool in a room whose temperature is a constant 0 \degree C.
By using the table below, find the equation for the temperature of the substance.
| \text{Minutes passed }(t) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Temperature }(T) | 145 | 87 | 52.2 | 31.32 | 18.792 |
According to the equation, what will the temperature be after 10 minutes, to two decimal places?
According to the model will the temperature of the substance ever reach 0 \degree C?
The temperature of another substance Q which also starts off at a temperature of 145 \degree C is modelled by T = 145 \times 0.9^{t}.
Which substance will cool more rapidly?
A colony of termites has moved onto a timber logging site, and the population, P, of the termites is given by P = 800e^{\frac{t}{4}}, where t is the time in weeks since the colony was established.
Find the population after 2 weeks to the nearest whole number.
Substitute P = 1600 into the equation to find the number of weeks it would take for the number of termites to reach 1600.
State whether the following statements are True or False:
As the population of termites gets larger, they increase at an increasing rate.
As the number of weeks pass, the population's increase slows down.
As the population of termites gets larger, the population's increase slows down.
As ther number of weeks pass, the population increases at an increasing rate.
Sketch the graph of P.
The weight, in grams, of a radioactive substance left after t years is given by W(t) = W_0 e^{- 0.018t} where W_0 is the initial weight of the substance.
What percentage of the substance is remaining after 5 years? Round your answer to two decimal places.
At what percentage is the substance decaying at per year?
What is the approximate half-life of the substance? Round your answer to two decimal places.
How long will it take for there to be 10\% of the substance remaining? Round your answer to two decimal places.
A fixed-rate investment generates a return of 6\% per annum, compounded annually. The value of the investment is modelled by A = P \left(1.06\right)^{t}, where P is the original investment.
Find the value of the investment after after 3\dfrac{1}{4} years if the original investment was \$200.
Maria purchased an artwork for \$2000 as an investment. At the end of each year, its value increases by 7\% of its value at the beginning of the year.
Find an equation to model the value, V, of the artwork t years after purchase.
Find the expected value 5 years after purchase to the nearest dollar.
A car originally valued at \$28\,000 is depreciated at the rate of 15\% per year. The salvage value, S, of the car after n years is given by S = 28\,000 \left(1 - \dfrac{15}{100}\right)^{n}.
Complete the following table. Round your answers to two decimal places.
| n | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|
| S |
Sketch the graph of S = 28\,000 \left(1 - \dfrac{15}{100}\right)^{n}.
What is the value of the car after 3 years?
After how many years does the value of the car drop down to an amount of \$6\,485.27?
The equation A = 12\,000 \times 1.15^{n} can be used to calculate the value, A, of a \$12\,000 investment at 15\% p.a., with interest compounded annually, after n years.
Calculate the value of the investment after 2 years.
Complete the table below:
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| A |
Sketch the graph of the function.
How many years, n, does it take for the balance to reach \$20\,988.08?
\$3000 is invested at 18\% p.a., with interest compounded annually.
Write the equation for the value, A, of the investment after n years.
Find the value of the investment after 6 years.
Complete the table below:
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| A |
Sketch the graph of A.
The equation A = 3000 \times 1.12^{n} can be used to calculate the value, A, of a \$3000 investment at 12\% p.a., with interest compounded annually, after n years.
Graph the function A = 3000 \times 1.12^{n} for 0 \leq n \leq 10.
Write down the equation for A if the interest was instead compounded semiannually?
Graph this new function for 0 \leq n \leq 10.
Write down the equation for A if interest was instead compounded quarterly?
Graph this new function for 0 \leq n \leq 10.
\$700 is invested at 4\% p.a. compound interest. The value of the money, A, after n years is given by A = 700 \left(1 + \dfrac{4}{100}\right)^{n}. The function has been graphed below:
Using the equation A = 700 \left(1 + \dfrac{4}{100}\right)^{n}, determine how much money would be in the account after 7 years, to the nearest dollar.
According to the graph, approximately how many years does it take for the balance to reach \$1120?
How much interest is earned during the 7th year?
Vanessa invests \$1000 into a term deposit, which is compounded at a rate, r, each year, as shown in the table:
| \text{Years passed }(n) | 0 | 1 | 2 |
|---|---|---|---|
| \text{Value of investment }(\$A) | 1000 | 1020 | 1040.40 |
Use the table to determine the value of r, leaving your answer as a decimal.
Find the rule for A, the value of the investment, in terms of n, the number of years after it is invested.
What will be the value of the investment after 6 years?
A new car purchased for \$38\,200 depreciates at a rate r\% each year. The value of the care for the first two years is shown in the table below:
| \text{Years passed }(n) | 0 | 1 | 2 |
|---|---|---|---|
| \text{Value of car }(\$A) | 38\,200 | 37\,818 | 37\,439.82 |
Find the value of r.
Write the rule for A, the value of the car n years after it is purchased.
Assuming the rate of depreciation remains constant, how much can the car be sold for after 6 years?
A new motorbike purchased for the same amount depreciates according to the model V = 38\,200 \times 0.97^{n}. Which vehicle depreciates more rapidly?
Do the graphs of the following functions illustrate exponential growth or exponential decay?
The population, P, of bacteria in a colony over time, t seconds, is given by P = e^{ 0.1 t}.
Sketch the graph of the population of bacteria over time for 0 \leq t \leq 10.
The growth rate per hour of a population of bacteria is 20\% of the current population.
At t = 0, the population is 4000. Sketch the graph of the population of bacteria over time for 0 \leq t \leq 10.
The proportion, P, of the current blue-green algae population to the initial population satisfies the equation P = e^{ 0.0009 t}, where t is the number of days since measurement began.
Find the number of days it takes the initial number of algae to double to the nearest two decimal places.
A country’s population t years after 2010 is given by P = 8\,837\,200 e^{ - 0.003 t }.
What was the country’s population in 2010?
Is the country’s population increasing or decreasing over time?
What is the country’s population in 2025? Round your answer to the nearest whole number.
How many whole years will it take until the population is half of what it was in 2010?
A model for a population, P, of foxes after t years is given by P \left( t \right) = 600 e^{ 0.05 t}.
Is the population increasing or decreasing over time?
How many foxes are there predicted to be after 2 years? Round your answer to the nearest whole number.
How many whole years will it take until there are approximately 900 foxes?
If the population continues to grow at this rate, approximately how long will it take for the population to double? Round answer to two decimal places.
What percentage increase is the population growing by each year? Round your answer to two decimal places.
The weight, in grams, of a radioactive substance left after t years is given by \\ W \left( t \right) = 250 e^{ - 0.015 t }.
What is the initial weight of the substance?
What percentage of the substance is remaining after 5 years? Round your answer to the nearest percent.
What is the approximate half-life of the substance? Round your answer to two decimal places.
Does the half life of the substance depend on its initial weight?
How long will it take for there to be 10\% of the substance remaining? Round your answer to two decimal places.
A model for a population, P, of birds vulnerable to extinction after t years is given by P \left( t \right) = 800 e^{ - 0.09 t }.
How many birds are there predicted to be after 3 years? Round your answer to the nearest whole number.
What percentage is the population decreasing by each year? Round your answer to two decimal places.
If the population continues to decay at this rate, how long will it take in years for there to be only 200 birds remaining? Round your answer to two decimal places.
The population of feral cats in a local council district has been monitored since January 2010.
The population P is modelled by P = 4000 e^{ - 0.0018 t }, where t is the number of years since the beginning of 2010.
State the initial population of feral cats.
State the continuous rate of change of the number of cats in this district.
Find the population at the beginning of 2015.
Find the number of years it will take for the initial population of cats to halve. Round your answer to one decimal place.
Yemen's population, t years after 2000, is modelled by P = 16.3 e^{ 0.0073 t}, where P is the number of millions of people.
State the population of Yemen in 2000 in millions of people.
State the continuous growth rate used in this model.
Find Yemen's population in 2009 in millions of people. Round your answer to one decimal place.
Find the number of whole years it will take for Yemen's population to reach 21 million.
Find the number of years it will take for Yemen's population to double. Round your answer to one decimal place.
When an object is heated to a certain temperature and then placed in a room of constant temperature 40 \degree C, the object starts to cool.
The temperature of the object, T, after t minutes is given by T = 40 + 1020 e^{ - 0.014 t }.
What is the initial temperature of the object when it is placed in the room?
Find the temperature of the object after 13 minutes to the nearest degree celsius.
Find the number of minutes it will take for the object to reach a temperature of 530 \degree C, to the nearest minute.
If the object is never removed from the room, what temperature will it eventually reach?
A local council has noticed that the rabbit population has increased rapidly, and begins to monitor their population.
They approximate that the number of rabbits is given by N \left( t \right) = \dfrac{B}{0.15 + e^{ - t }}, where t is the number of months after they started monitoring.
Given that they initially record 53 rabbits in the area, find the value of B. Round your answer to two decimal places.
What would the council expect the rabbit population to reach 1 month after they started monitoring, to the nearest whole number?
What would the council expect the rabbit population to reach 1 year after they started monitoring, to the nearest whole number.
The height, in metres, of a particular tree after x years is modelled by f \left( x \right) = \dfrac{40}{1 + 47.3 e^{ - 0.21 x }}
Complete the following table. Round all values to one decimal place.
| x | 10 | 20 | 30 | 40 | 50 | 60 | 70 |
|---|---|---|---|---|---|---|---|
| f \left( x \right) |
From the table, what appears to be the maximum height of the tree?
The graph of y = f \left( x \right) is shown:
What is the equation of the horizontal asymptote as x tends to infinity?
What is the significance of this asymptote?
How long does it take for the height of the tree to reach 21 m? Round your answer to one decimal place.
The likelihood of a person having a certain disease increases with their age.
The proportion of x year old people who have the disease is modelled by f \left( x \right) = \dfrac{0.8}{1 + 272 e^{ - 0.122 x }}So f(10) equals the proportion of 10 year old people who have the disease.
Find the value of f \left( 35 \right) to three decimal places.
Describe the meaning of your result from part (a).
Find the age at which the likelihood of having the disease equals 50\%. Round your answer to the nearest whole number.
The population of an organism in an area is modelled by the function G \left( x \right) = \dfrac{250\,000}{100 + 2400 e^{ - x }}where x is the number of decades since the organism started growing.
Use a graphics calculator to draw the function, for 0 \leq x \leq 8, and 0 \leq y \leq 2500, and estimate the value of G\left( 2 \right) from the graph, to the nearest ten.
Find the value of G \left( 2 \right) using the equation, in order to accurately find the population after 2 decades. Round your answer to the nearest whole number.
By referring to the graph on your calculator, estimate the number of decades required for the population to reach 1000. Round your answer to one decimal place.
Find the number of decades required for the population to reach 1000 by solving the equation algebraically using your calculator. Round your answer to four decimal places.
A patient is administered a dose of a pain-killing drug. The number of units of the drug (D) in the blood system over a period of time (t hours) can be modelled by the following function D \left( t \right) = 6.8 t e^{ - 0.5 t }.
Use technology to draw the function for 0 < t < 12.
How many units of the drug are in the system after 1 hour? Round your answer to two decimal places.
What is the maximum number of units of the drug in the system? Round your answer to the nearest whole number of units.
The drug is usually effective when there is 2 units of the drug in the system. How long, in hours, after administering the drug does it becomes effective? Round your answer to two decimal places.
A new dose will be given when the level drops below 2 units once more. If the first dose was given at 10 AM, when can a second dose be given?