Standard +0.3 Part (a) requires direct substitution into a given formula and evaluating sin(26.565°), which is straightforward calculator work. Part (b) involves rearranging to isolate the sine function, using inverse sine, and solving for t with consideration of the periodic nature—standard A-level technique but requires careful handling of the general solution. This is slightly easier than average due to the formula being provided and the methods being well-practiced, though the real-world context and degree mode add minor complexity.
8. The length of the daylight, \(D ( t )\) in a town in Sweden can be modelled using the equation
$$D ( t ) = 12 + 9 \sin \left( \frac { 360 t } { 365 } - 63.435 \right) \quad 0 \leq t \leq 365$$
where \(t\) is the number of days into the year, and the argument of \(\sin x\) is in degrees
a. Find the number of daylight hours after 90 days in that year.
b. Find the values of \(t\) when \(D ( t ) = 17\), giving your answers to the nearest integer. (Solutions based entirely on graphical or numerical methods are not acceptable)
\(t = 98.5 = 99\) days and \(t = 212.598 = 213\) days
M1
Using the correct order to find a second value of \(t\)
A1
\(\sin\left(\frac{360t}{365} - 63.435\right)^0 = \frac{5}{9}\). Using the correct order to find one correct value of \(t\).
M1
For using \(D = 17\) and proceeding to \(\sin\left(\frac{360t}{365} - 63.435\right)^0 = k\) where \(
k
### Part a:
$15.85 \ldots$ hours | B1 | Finding $D(90) = 15.85 \ldots$ hours
### Part b:
$t = 99$ or $213$ to the nearest integer | A1 | $t = 98.5 = 99$ days and $t = 212.598 = 213$ days
| M1 | Using the correct order to find a second value of $t$
| A1 | $\sin\left(\frac{360t}{365} - 63.435\right)^0 = \frac{5}{9}$. Using the correct order to find one correct value of $t$.
| M1 | For using $D = 17$ and proceeding to $\sin\left(\frac{360t}{365} - 63.435\right)^0 = k$ where $|k| \leq 1$
8. The length of the daylight, $D ( t )$ in a town in Sweden can be modelled using the equation
$$D ( t ) = 12 + 9 \sin \left( \frac { 360 t } { 365 } - 63.435 \right) \quad 0 \leq t \leq 365$$
where $t$ is the number of days into the year, and the argument of $\sin x$ is in degrees\\
a. Find the number of daylight hours after 90 days in that year.\\
b. Find the values of $t$ when $D ( t ) = 17$, giving your answers to the nearest integer. (Solutions based entirely on graphical or numerical methods are not acceptable)\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q8 [5]}}