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The length of the day in Lagos, Nigeria (in minutes),
t days since the beginning of the year can be estimated by this formula:

L(t)=727+22 sin((2pi(t-80.75))/(365)).
When is the longest day of the year? Give an exact answer.
days since the beginning of the year

The length of the day in Lagos, Nigeria (in minutes), $t$ days since the beginning of the year can be estimated by this formula: $\newline$ $L(t)=727+22 \sin \left(\frac{2 \pi(t-80.75)}{365}\right) .$ $\newline$ When is the longest day of the year? Give an exact answer. $\newline$ $\square$ days since the beginning of the year

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Find trigonometric functions using a calculator

Full solution

Q. The length of the day in Lagos, Nigeria (in minutes),

t

days since the beginning of the year can be estimated by this formula:

\newline

L(t)=727+22 \sin \left(\frac{2 \pi(t-80.75)}{365}\right) .

\newline

When is the longest day of the year? Give an exact answer.

\newline

\square

days since the beginning of the year

Longest day calculation: The longest day of the year occurs when the sine function in the formula reaches its maximum value, which is $1$ .
Setting up sine function: To find when the sine function is equal to $1$ , we set the inside of the sine function equal to $\pi/2$ , because $\sin(\pi/2) = 1$ .
Solving for t: So we solve the equation $(2\pi(t-80.75))/365 = \pi/2$ .
Multiplying and dividing: Multiply both sides by $365$ to get rid of the denominator: $2\pi(t-80.75) = \left(\frac{\pi}{2}\right) \times 365$ .
Calculating right side: Divide both sides by $2\pi$ to solve for $t$ : $t-80.75 = \frac{365}{4\pi}$ .
Adding to find $t$ : Now, calculate the right side: $\frac{365}{4\pi} \approx 29.09$ .
Final calculation: Add $80.75$ to both sides to find $t$ : $t \approx 29.09 + 80.75$ .
Final calculation: Add $80.75$ to both sides to find $t$ : $t \approx 29.09 + 80.75$ .Calculate the sum: $t \approx 109.84$ .

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