Answer to Question #281047 in Trigonometry for Aliyah

Question

7.Two ships leave a dock at the same time. One sails northeast at the rate of 8.5 miles an hour, the other sails north at the rate of 10 miles an hour. How far apart are they at the end of 2 hours?

8. If the slower ship in the preceding exercise leaves at noon, and the other at 1 p.m., how far apart are they at 2 p.m.? 9. The diagonals of a parallelogram are 7 inches and 9 inches respectively; they intersect at an angle of 52°. Find the sides of the parallelogram.

10. A military observer notes two enemy batteries which sub tend, at his observation post, an angle of 40°. The interval between the flash and the report of a gun is 5 seconds for one battery, and 4 seconds for the other. If the velocity of sound is 1140 feet a second, how far apart are the batteries?

Answer 1

7.

$$s_1=2(8.5)=17(mi)$$

$$s_2=2(10)=20(mi)$$

$$s=\sqrt{s_1^2+s_2^2-2s_1s_2\cos45\degree}$$

$$=\sqrt{17^2+20^2-2(17)(20)(\dfrac{\sqrt{2}}{2})}(mi)$$

$$=\sqrt{689-340\sqrt{2}}(mi)\approx14.428(mi)$$

8.

$$s_1=2(8.5)=17(mi)$$

$$s_2=2(10)=20(mi)$$

$$s=\sqrt{s_1^2+s_2^2-2s_1s_2\cos45\degree}$$

$$=\sqrt{17^2+10^2-2(17)(10)(\dfrac{\sqrt{2}}{2})}(mi)$$

$$=\sqrt{389-170\sqrt{2}}(mi)\approx12.189(mi)$$

9.

$$BD=7\ in,AC=9\ in$$

$$\angle AOB=52\degree$$

$$\angle AOB=180\degree-52\degree=128\degree$$

$$AO=OC=4.5\ in, BO=OD=3.5\ in$$

The Law of Cosines

$$a=\sqrt{(4.5)^2+(3.5)^2-2(3.5)(4.5)\cos(128\degree)}$$

$$\approx7.2037(in)$$

$$b=\sqrt{(4.5)^2+(3.5)^2-2(3.5)(4.5)\cos(52\degree)}$$

$$\approx3.6203(in)$$

10.

$$s_1=5(1140)=5700(ft)$$

$$s_2=4(1140)=4560(ft)$$

$$s=\sqrt{s_1^2+s_2^2-2s_1s_2\cos40\degree}$$

$$=\sqrt{5700^2+4560^2-2(5700)(4560)\cos40\degree}(ft)$$

$$\approx3669(ft)$$