The amount of water, A, in thousands of litres, available in a water tank located on a farm fluctuates in a yearly cycle and can be modelled by the function A(t) = a sin (kt) + b, where t is the elapsed time, in weeks, since the start of the year. The amount of water available in the tank on week 6 is 24 thousand litres and on week 31 is 9.2 thousand litres. (a) Find the value of k, in degrees, assuming there are 52 whole weeks in a year. (b) Set up a pair of equations to find the value of a and the value of b. Give your answers correct to the nearest integer. (c) Hence find the amount of water available in the tank in week 42.
The amount of water, A, in thousands of litres, available in a water tank located on a farm
fluctuates in a yearly cycle and can be modelled by the function
A(t) = a sin (kt) + b,
where t is the elapsed time, in weeks, since the start of the year.
The amount of water available in the tank on week 6 is 24 thousand litres and on week 31 is
9.2 thousand litres.
(a) Find the value of k, in degrees, assuming there are 52 whole weeks in a year.
(b) Set up a pair of equations to find the value of a and the value of b. Give your answers
correct to the nearest integer.
(c) Hence find the amount of water available in the tank in week 42.
+
(d) Draw the graph of y = A(t) on the grid below, for one full year, indicating clearly
the minimum and maximum points.
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