Make t the subject for the following equations:
v = u - a \ln t
r = \dfrac{u}{\ln t} - b
A plane takes off from an airport at sea level and its altitude h, in metres, t minutes after taking off, is given by h = 600 \ln \left(t + 1\right).
Write an expression for the rate at which the plane ascends, exactly t minutes after taking off.
Hence, find the rate of ascent at exactly 4 minutes after take off.
Determine whether the plane is ascending at an increasing or decreasing rate.
Bart releases his new book on the 6th of January. The number of sales of the book per day since its release is displayed in the given graph:
From the first day of release onwards, the book enjoys a steady growth in sales. This growth is roughly modelled by the equation: f \left( t \right) = 1483 + 93 \ln t where t is the number of days since the book's release.
Find the expected number of sales on the 8th day after its release.
The book's publisher tells Bart that the number of book sales on the 8th day is actually 1706. By how much was the estimate in previous part different to the actual number of sales on the 8th day?
Beth uploads a new video to her popular website. The number of people viewing the video (in thousands) is roughly modelled by the equation: V = - 180 + 57 \ln x where x is the number of minutes since the video was upload.
The following table shows the number of people who have viewed the video (in thousands) at certain intervals (in minutes) since she completed the upload:
| Minutes since upload | Number of views (in thousands) |
|---|---|
| 30 | 13.9 |
| 45 | 37.0 |
| 60 | 53.4 |
| 75 | 66.1 |
| 90 | 76.5 |
Find V, after it has been online for 80 minutes. Round your answer correct to one decimal place.
What assumption must we make in order to estimate the number of people viewing the video after the video has been online for 90 minutes?
That the increase in the number of viewers does not stop
That the increase in of viewers eventually stops.
That the increase in viewers continues to be logarithmic.
That the increase in viewers continues to be linear.
Given that the assumption of part (b) is true, estimate the number of viewers (in thousands) when the video has been online for 2 hours. Round your answer to one decimal place.
After 2 hours, Beth finds that the number of views (in thousands) for her video is 90.5. By how much was the estimate in part (c) off the actual number of views after 2 hours? Round your answer to one decimal place.
Before conservationists release two different species of endangered birds back into the wild, they introduce some of them into a large enclosure and track their numbers over 20 months.
The populations of species A and B, t months after being in the enclosure, are given by the following functions:
- Species A: A \left( t \right) = 17 \ln \left( 8 t + 5\right)
- Species B: B \left( t \right) = 17 \ln \left( 6 t + 25\right)
State the domain of both functions.
After 1 year, at what rate is the population of species A increasing?
Calculate the number of whole months, t, in the enclosure, for the populations of species A and B to reach the same level.
At the time when the populations of the two species are the same, find the exact rate of increase in the population of:
Species A
Species B
Hence, state which species will have a greater population after 20 months.
An object has a displacement function s \left( t \right) = t - 4 \ln \left( 8 t + 1\right) where s is in metres and t is in seconds.
Find the initial position of the object.
Find the equation of the velocity function, v \left(t\right).
State the time, t, that the object changes direction.
State how far the object travels in the first 5 seconds. Round your answer to two decimal places.
At what time does the object return to the origin? Round your answer to two decimal places.
A new airline wants to forecast its predicted growth in passenger numbers from now until the end of the decade.
In the current month \left( t = 0 \right), the airline services 3400 passengers. The forecasters use the following function to model the number of passengers t months from the current month, for t \geq 0 and some constants m and k:
P \left( t \right) = k \ln \left(t + 1\right) + mState the value and explain the meaning of the constant m.
Find k, if 4 months later the airline services 3641 passengers. Round your answer to the nearest whole number.
Find t, the whole number of months it will take for the airline to reach 3745 passengers.
Hence, find the initial rate P \rq \left( 0 \right), at which the number of passengers is increasing.
Find the number of months, t, for the rate of increase to be half the initial rate of increase.
Luigi's farm currently produces 10.1 tonnes of barley annually. Over an extended period of drought, he has found that the productivity of his land is decreasing but the rate of decrease is slowing down.
He decides that he will keep his barley farm until annual productivity reaches 0, so he uses the following logarithmic function to model the annual productivity of his land, t years from now:
P \left( t \right) = A + k \ln \left(t + 1\right)Note that P \left( t \right) is the annual productivity in tonnes after t years, and P \left( 0 \right) is the annual rate of productivity at the start of the model.
Find A.
Hence, find the value of k in the model if the annual productivity drops to 7 tonnes after one year. Round your answer to one decimal place.
Find the rate at which the annual productivity changes 4 years from now, correct to one decimal place.
Find t, the number of years from now that Luigi will sell his farm. Assume that he only sells at the end of the year.
Determine the rate at which the annual productivity is decreasing at the end of the last year that Luigi runs the farm. Round your answer to two decimal places.
Find the equation of the curve f \left( x \right), given the derivative function and a point on the curve:
f' \left( x \right) = \dfrac{5}{5 x - 4}, and the point \left(3, \ln 11\right).
f' \left( x \right) = \dfrac{6}{2 x + 5} , and the point \left(1, \ln 49\right).
Consider the functions y = \ln x and y = \ln \left( 4 x - 9\right).
Find the x-coordinate of the point of intersection.
Find the exact area bounded by the two curves and the x-axis.
A circus tent is 7 \text{ m} high and has a radius of 6 \text{ m}. The equation to describe the curved roof of the tent is y = \dfrac{7}{x + 1}, as shown in the diagram:
Calculate the exact cross-sectional area of the tent.
Consider the given graph of y = \dfrac{1}{x}.
Find the value of k such that the area A is 1 \text{ units}^{2}.
Consider the given graph of y = \dfrac{2}{x}.
Find the value of k such that the areas A and B are equal.
The cross-section of a satellite dish can be estimated by the area bound by the hyperbola y = \dfrac{32}{x} \,and the line y = - x + 12 , as shown in the diagram:
Find the x-values of the points of intersection of the hyperbola and the line.
Hence determine the cross-sectional area of the satellite dish, correct to two decimal places.
The given graph shows the function f \left( x \right) = \dfrac{3}{x + 1}:
Find an estimate for the area between the function and the x-axis for 1 \leq x \leq 6 using the right-endpoint approximation and 3 rectangles. Round your answer to three decimal places.
Find an estimate for the area between the function and the x-axis for 1 \leq x \leq 6 using the right-endpoint approximation and 5 rectangles. Round your answer to three decimal places.
Using technology, find an estimate for the area between the function and the \\x-axis for 1 \leq x \leq 6 using the right-endpoint approximation method and 60 rectangles. Round your answer to four decimal places.
Find the exact area between the function and the x-axis for 1 \leq x \leq 6.
Calculate the percentage error of the approximation in part (c) compared to the exact area. Round your percentage to two decimal places.
A particle moves in a straight line so that after t seconds (t \geq 0) its velocity v is given by:v = \dfrac{3}{1 + t} - t + 1 \text{ m/s} The displacement of the particle from the origin is given by x metres.
If the particle is initially at the origin, find the displacement as a function of time t.
Solve for the time, t, at which the particle is stationary.
How far does the particle travel in the first 4 seconds? Round your answer to two decimal places.
The acceleration function of an object where a is in \text{m/s}^{2} and t is in seconds is given by:a \left( t \right) = - \dfrac{12}{\left( 2 t + 1\right)^{2}}The object is initially stationary at the origin.
Find the velocity function, v \left( t \right).
Find the displacement function, s \left( t \right).
How far does the object travel in the first 4 seconds?
Find the object's speed at t = 1.