The equation for the population at time t is given by Q = N e^{ 2 t}. Make t the subject of the equation.
Make P the subject of the equation M = 4 - e^{P}.
Make t the subject for the following equations:
v = u - a \ln t
r = \dfrac{u}{\ln t} - b
Make k the subject of the equation P = A + B e^{ - k t }.
Solve for x in the following equations:
y = A \left(1 + e^{ - B x }\right) - C
T = \dfrac{R}{1 + a e^{ - n x }}
Consider the curve y = e^{x} - 9.
Find the exact value of the x-intercept.
Find the exact area between the curve y = e^{x} - 9 and the x-axis, from the lines x = 0 to x = 3.
Consider the functions f \left( x \right) = 3 - e^{x} and g \left( x \right) = e^{x} + 1 .
Find the x-coordinate of the point of intersection.
Calculate the exact area bound between the two curves and the line x = \ln 3.
Under certain climatic conditions the proportion, P , of the current blue-green algae population compared to the initial population satisfies the equation P = e^{ 0.007 t}, where t is measured in days from when measurement began.
Solve for t, 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 = 9\,008\,900 e^{ - 0.006 t }
State the country’s population in 2010.
Is the country’s population increasing or decreasing over time?
How many whole years will it take until the population is half of that in 2010?
In a particular geographic location the population of a particular species of predator and the population of a particular species of prey is monitored each month. The population of the species of predator and prey where t is the number of months are given by:
Species of predator: A = 6000 e^{ 0.01 t}
Species of prey: B = 12\,000 e^{ - 0.02 t }
Solve for the time t at which the two populations are equal, correct to one decimal place.
The voltage V in volts across an electrical component where A is the initial voltage and t is the time in years, decays according to the equation:V = A e^{ - \frac{1}{2} t }
Find the value of t such that V is half of the initial voltage. Round your answer to two decimal places.
The spread of a virus through a city where N is the number of people infected by the virus after t days, is modelled by the function:N = \dfrac{15\,000}{1 + 100 e^{ - 0.5 t }}
Calculate the number of people that will have been infected after 3 days.
How many whole days will it take for at least 4000 people to be infected with the virus?
100 \text{ g} of sugar is placed in a container of water and begins to dissolve. After t hours, the amount A \text{ g} of undissolved sugar remaining is given by: A = 100 e^{ - k t }
After 5 hours, 6.4 \text{ g} of undissolved sugar remains.
Solve for the exact value of k.
Solve for the number of hours t it takes for 20 \text{ g} of sugar to remain undissolved, correct to two decimal places.
In water, visibility is a measure of the furthest distance at which an object can still be seen. At a depth of d metres below the surface, the proportion of visibility relative to the visibility at surface level, where k is a positive constant, is given by:P = e^{ - k d }
The visibility at a depth of 60 \text{ m} below the surface is 3 tenths of the visibility at the surface.
Find the exact value of k.
Determine the depth, d , at which the visibility is \dfrac{1}{5} of the visibility at the surface, correct to two decimal places.
The remains of a human body can be dated by measuring the proportion of radiocarbon in tooth enamel. The proportion of radiocarbon, A, remaining t years after a human passes away is given by: A = C e^{ - k t } where k is a positive constant.
Solve for the exact value of k if the amount of radiocarbon present is halved every 5594 years.
For a particular corpse, state the number of years since the person passed away, if the amount of radiocarbon present is exactly 25\% of the original amount at death.
A compound substance breaks down into its elements at a rate such that the amount of the compound remaining is given by: N = A e^{ - k t }
where N is the amount of the compound remaining after t seconds, and A and k are constants.
If 300 \text{ g} of the compound decomposes down to 250 \text{ g} in 50 seconds, find the exact values of A and k.
Find the amount of the compound remaining, in grams, after 2.75 minutes, correct to two decimal places.
According to this model, will the compound decompose completely? Explain your answer.
Oceanographers wanting to map the ocean floor drop a heavy metal ball fitted with a GPS from a helicopter into the sea below. When it hits the water, the velocity and acceleration, where t is time in seconds, are given by:
Velocity: v = B e^{ - k t }
Acceleration: a = - k B e^{ - k t }
When the ball first hits the water, at t = 0, it is falling with velocity 22 \text{ m/s}. After 3 seconds, its velocity has dropped to 17.6 \text{ m/s}.
Solve for the constant B.
Find the exact value of k.
Find the velocity of the ball after 7 seconds, correct to two decimal places.
Find the acceleration of the ball after 7 seconds, correct to two decimal places.
According to this model, would it be possible for the metal ball to come to rest before hitting the ocean floor (assuming no other external forces)?
The likelihood of a person having a certain disease increases with their age. The proportion of people of x years of age, who have the disease is modelled by:f \left( x \right) = \dfrac{0.9}{1 + 271 e^{ - 0.124 x }}
Find the value of f \left( 25 \right), correct to three decimal places.
Interpret the answer to part (a).
Find the age at which the likelihood of having the disease equals 50\%.
The population of an organism in an area is modelled by the logistic function:G \left( x \right) = \dfrac{M G_0}{G_0 + \left(M - G_0\right) e^{ - k M x }}where G_0 is the initial population, M is the maximum possible size of the population, and k is a positive constant.
Suppose G_0 = 100, M = 2500, k = 0.0004 and x is time in decades.
Use technology to graph the function, using 0 \leq x \lt 8, and 0 \leq y \lt 2500. Estimate the value of G \left( 3 \right) from the graph to the nearest ten.
Find the value of G \left( 3 \right) algebraically to find the population after 3 decades.
By referring to the graph, estimate the number of decades required for the population to reach 1000. Round your answer to one decimal place.
Solve algebraically, to get a value closer to the exact one, for the number of decades at at which the population is 1000. Round your answer to four decimal places.
When a nuclear reactor caught fire, residents of the local town were exposed to 193 units of iodine-131, a radioactive substance. After initial exposure, the mass N of iodine-131 present after t days where k and B are constants was given by:N = B e^{ - k t }
Iodine-131 has a half life of 8 days. Solve for B and the exact value of k.
The safe level of human exposure to iodine-131 is only 0.8\% of the initial amount that residents were exposed to. Determine the number of days it will take for amount of iodine-131 to drop to this safe level, to the nearest whole number.
Quiana takes a block of chocolate out of a refrigerator set to - 2 \degree \text{C} at 8:18 pm and leaves it on the kitchen bench, where the temperature is 28 \degree \text{C}. At 8:24 pm, the block of chocolate has a temperature of 23 \degree \text{C} .
By Newton's Law of Cooling, the temperature of the block of chocolate, T \left( t \right), at time t can be calculated by the following formula: T \left( t \right) = R + \left(A - R\right) e^{ - k t }
where R is the temperature of the room, A is the temperature of the block of chocolate and k is a constant.
Determine a simplified form of this equation for this case.
Find the value of k, correct to four decimal places.
Find the temperature of the block of chocolate, T, at 8:37 pm correct to two decimal places.
The population of China (in millions), t years after 1980, is modelled by C = 991 e^{ 0.011 t}.
The population of India (in millions), t years after 1980 is modelled by I = 680 e^{ 0.024 t}.
After how many years will India's population first exceed China's population. Round your answer to the nearest whole number.
Find the time, t, at which the instantaneous growth rate of India's population is double that of China. Round your answer to one decimal place.
One model that the World Bank uses to predict the number of individuals worldwide who will have internet access is N = A e^{ k t} + 1300
where N is the number of individuals, in millions, who will be connected to the internet, t years from now. It is estimated that 2900 million people will have internet access 1 year from now, and that this will increase to 3220 million people 2 years from now.
Solve for the exact values of A and k.
Describe the rate at which the number of individuals with internet access is changing.
At the site of a nuclear accident, the amounts of two radioactive substances are monitored. The amount of radioactive substance E and F remaining are modelled by:
Substance E: A = 68 e^{ - 0.02 t }
Substance F: B = 39 e^{ - 0.003 t }
where t is time in years, and A and B are amounts in micrograms.
Find the instantaneous rate of decay in terms of t for substance E.
Find the instantaneous rate of decay in terms of t for substance F.
Solve for the time, t, at which the instantaneous decay rates are equal. Round your answer to one decimal place.
A lost dog is eventually picked up by a traveller at a remote outpost. She finds that the dog is infested with fleas, and knows that without intervention, the number of fleas will be \\P = B e^{ k t} and increase at a rate of R = k B e^{ k t}, where B and k are constants, and t is the time in hours.
The traveller approximates about 170 fleas on the dog’s body, and decides to start driving to the nearest vet which is k days’ drive away. After 7 hours, she approximates that the number of fleas has increased to 200.
Find the exact value of k.
When the traveller initially picked up the dog, at what rate were the number of fleas increasing? Round your answer correct to two decimal places.
The traveller thinks treatment will be ineffective once the number of fleas reaches 1020. According to the model, how many hours will it take to reach this point? Round your answer to two decimal places.
Will the traveller make it to the vet in time for the treatment to be effective?
Coal is extracted from a mine at a rate that is proportional to the amount of coal, A, remaining in the mine after time t years.
The equation for the rate at which the coal is mined is \dfrac{dA}{dt} = - k A, where k is a positive constant
The more coal that is extracted from the mine, the slower or faster the remaining coal is extracted? Explain your answer.
Show that the solution to the equation \dfrac{dA}{dt} = - k A is A = C e^{ - k t }, for some constant C.
After 8 years, 50\% of the initial amount of coal has been mined and removed. Solve for the value of k.
The mine is to be shut down when there is only 19\% of the original amount remaining. For how many years, t, will the mine be open?
A charged particle moves back and forth about the fixed point x = 0 (called the origin). Its position, x \text{ cm} from the origin, after t seconds is given by the equation:
x = \sin \left( \pi e^{ 2 t}\right)Find the particle's initial position.
Find an expression for the velocity of the particle , v \left( t \right) = x' \left( t \right).
Find the exact times for the first and second occasion that the particle comes to a stop.
Describe the position of the particle when it first comes to a stop, v \left( t \right) = 0.
Describe the position of the particle when it comes to a stop for the second time.
A rechargeable battery has a voltage of 9 \text{ V} when fully charged. When the battery is used to run an electronic toy, the voltage, V volts, remains at 9 \text{ V} for 30 minutes and then it decreases instantaneously, at a rate modelled by:
\dfrac{d V}{d t} = - 0.3 e^{ - 0.03 t}Find the change in the battery voltage after the toy has been in use for 45 minutes. Round your answer to two decimal places.
Find the function V, the battery voltage after the toy has been in use for t minutes when t \geq 30.
Find the number of whole minutes the battery can be used to run this toy, if a minimum voltage of 7 \text{ V} is required.
Consider the function f \left( x \right) = 2^{x}.
Express the function in the form f \left( x \right) = e^{ k x}.
Hence, state the equation of the derivative of f \left( x \right).
Consider the function f \left( x \right) = 50 \times \left(1.8\right)^{x}.
Express the function in the form f \left( x \right) = 50 e^{ k x}.
Hence, state the equation of the derivative of f \left( x \right).
Consider the function y = 7^{x}.
Rewrite the function in terms of natural base e.
Hence determine y'. Express the derivative in terms of the base 7.
Hence determine the exact gradient of the tangent of the curve at x = 1.
Differentiate the following functions:
y = 5^{x}
y = 2 \left(7^{x}\right)
y = - 4^{x} + 3
y = x^{2} \left(5^{x}\right)
y = \dfrac{x^{3} - 1}{2^{x}}
y = 2^{ - t }
y = 5^{2 - 4 x} + 3
P \left( t \right) = 2^{ 3 t}
A population of wombats increases according to the model P \left( t \right) = 1800 \times \left(1.05\right)^{t} where t is measured in years.
Express the function in the form P \left( t \right) = 1800 e^{ k t}.
Determine the rate of change of the population after 3 years.
Find the value of t when the rate of change is 150 wombats per year. Round your answer to the nearest year.
A population of fish increases according to the model P \left( t \right) = 3500 \times \left(0.85\right)^{t} where t is measured in years.
Express the function in the form P \left( t \right) = 3500 e^{ k t}.
Determine the rate of change of the population after 18 months.
Find the value of t when the population is decreasing at a rate of 100 fish per year. Round your answer to two decimal places.
Find the value of t when the population is half the amount of the initial population. Round your answer to two decimal places.
Consider the function f \left( x \right) = 10 \times 4^{x}.
Express the function in the form f \left( x \right) = 10 e^{ k x}.
Hence find \int f \left( x \right) dx.
Consider the function f \left( x \right) = 2 \times 3^{x}.
Express the function in the form f \left( x \right) = 2 e^{ k x}.
Hence, find the exact area bounded by f \left( x \right), the x-axis and the lines x = 0 and x = 1.