find pq.\nwrite your answer as an integer or as a decimal rounded to the nearest tenth.\npq = \nsubmit

Explanation:

Step1: Identify triangle type

It's a right - triangle ( \triangle PQR ) with ( \angle Q = 90^{\circ} ), hypotenuse ( PR=\sqrt{94}), and ( \angle R = 25^{\circ} ). We want to find ( PQ ), which is the side opposite to ( \angle R ).

Step2: Use sine function

In a right - triangle, the sine of an angle ( \theta ) is defined as ( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} ). Here, ( \theta = 25^{\circ} ), opposite side to ( \theta ) is ( PQ ), and hypotenuse is ( PR = \sqrt{94}\approx9.7 ) (first, calculate the approximate value of ( \sqrt{94} ): ( 9^2 = 81 ), ( 10^2=100 ), ( 9.7^2=(10 - 0.3)^2 = 100-6 + 0.09 = 94.09\approx94 )).

Using the sine formula: ( \sin(25^{\circ})=\frac{PQ}{PR} ), so ( PQ=PR\times\sin(25^{\circ}) ).

We know that ( \sin(25^{\circ})\approx0.4226 ) and ( PR = \sqrt{94}\approx9.7 ).

Then ( PQ\approx9.7\times0.4226\approx4.1 )

Answer:

( 4.1 )