Question

if $\angle e$ measures $45^\circ$, $\angle f$ measures $98^\circ$, and $e$ is 5 feet, then find $f$ using the law of sines. round your answer to the nearest foot.\
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options: 4 feet, 5 feet, 6 feet, 7 feet\
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question 15 (multiple choice worth 1 points)\
(05.03 mc)\
the picture below shows a box sliding down a ramp:\
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what is the distance, in feet, that the box has to travel to move from point a to point c?\
options: $12\cos 65^\circ$, $\frac{12}{\sin 65^\circ}$, $12\sin 65^\circ$, $\frac{12}{\cos 65^\circ}$

Explanation:

First Problem

Step1: Recall Law of Sines

$\frac{f}{\sin\angle F} = \frac{e}{\sin\angle E}$

Step2: Substitute given values

$\frac{f}{\sin98^\circ} = \frac{5}{\sin45^\circ}$

Step3: Solve for f

$f = \frac{5\times\sin98^\circ}{\sin45^\circ}$

Step4: Calculate values

$\sin98^\circ\approx0.9903$, $\sin45^\circ\approx0.7071$
$f\approx\frac{5\times0.9903}{0.7071}\approx6.96$

Step5: Round to nearest foot

$f\approx7$

Step1: Identify right triangle parts

In right $\triangle ABC$, $\angle B=90^\circ$, $AB=12$ ft, $\angle C=65^\circ$, $AC$ is the hypotenuse (distance from A to C).

Step2: Use sine definition

$\sin\angle C = \frac{AB}{AC}$

Step3: Rearrange to solve for AC

$AC = \frac{AB}{\sin\angle C} = \frac{12}{\sin65^\circ}$

Answer:

7 feet

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Second Problem (Question 15)