Question

\i think i can make it. ill just jump,\ says denis, flashing a grin as he looks to the neighboring rooftop (as shown below).

\wait!\ vera exclaims. \first lets find the distance! we know were 4.7 m apart. i see a 40° angle between you and your landing point. from your point of view, theres a 105° angle between me and your landing point. all we need to do is...\

denis jumps.

\...use some trig,\ vera gasps.

image: triangle with denis, vera, neighboring rooftop. denis and vera are 4.7m apart. angle at vera: 40°, angle at denis: 105°, distance from denis to rooftop is \?\.

how far did denis need to jump to reach the rooftop?
do not round during your calculations. round your final answer to the nearest tenth of a meter.

______ m

Explanation:

Step1: Find the third angle of the triangle

The sum of angles in a triangle is \(180^\circ\). Given two angles: \(105^\circ\) (at Denis) and \(40^\circ\) (at Vera), the third angle \(C\) is \(180 - 105 - 40 = 35^\circ\).

Step2: Apply the Law of Sines

The Law of Sines states \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let \(a\) be the distance Denis jumps (opposite \(40^\circ\)), \(b = 4.7\) m (opposite \(35^\circ\)). So \(\frac{a}{\sin 40^\circ}=\frac{4.7}{\sin 35^\circ}\).

Step3: Solve for \(a\)

\(a=\frac{4.7\times\sin 40^\circ}{\sin 35^\circ}\). Calculate \(\sin 40^\circ\approx0.6428\), \(\sin 35^\circ\approx0.5736\). Then \(a=\frac{4.7\times0.6428}{0.5736}\approx\frac{3.02116}{0.5736}\approx5.267\). Round to nearest tenth: \(5.3\).

Answer:

\(5.3\)