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Bookwork code: D21
Use the information below to work out the closest distance that a cinema can put seats to the screen.
Give your answer to $1 \mathrm{~d} . p$.
Safety rules say that the angle of elevation from a customer's eyes to the top of the screen must be no more than $31^{\circ}$.
The top of the cinema screen is $7.9 \mathrm{~m}$ above the floor.
Customers' eyes are $1.2 \mathrm{~m}$ above the floor when they are sat on a seat.
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Answer
\(\boxed{11.2}\) meters is the closest distance that a cinema can put seats to the screen.
Step 1 ：Given the angle of elevation $31^\circ$, the height of the screen $7.9 \mathrm{~m}$, and the height of a customer's eyes when seated $1.2 \mathrm{~m}$.
Step 2 ：Calculate the height difference between the screen and the customer's eyes: $7.9 - 1.2 = 6.7 \mathrm{~m}$.
Step 3 ：Use the tangent function to find the distance: $\tan(31^\circ) = \frac{6.7}{x}$.
Step 4 ：Solve for x: $x = \frac{6.7}{\tan(31^\circ)} \approx 11.2 \mathrm{~m}$.
Step 5 ：\(\boxed{11.2}\) meters is the closest distance that a cinema can put seats to the screen.
