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QuestionFind the distance from City A to City C, given bearings and travel times. Round to the nearest mile.

Studdy Solution

STEP 1

Assumptions1. The bearing from City A to City B is 48∘48^{\circ} \mathrm{}. . The bearing from City B to City C is 42∘42^{\circ} \mathrm{}.
3. The speed of the automobile is 55mph55 \mathrm{mph}.
4. The time taken to drive from City A to City B is 1.81.8 hours.
5. The time taken to drive from City B to City C is 1.61.6 hours.
6. We are asked to find the distance from City A to City C.
7. We neglect the curvature of the earth, so we can use plane geometry.

STEP 2

First, we need to find the distance between City A and City B. We can do this by multiplying the speed of the automobile by the time taken to drive from City A to City B.
DistanceAB=SpeedABtimesTimeABDistance_{AB} = Speed_{AB} \\times Time_{AB}

STEP 3

Now, plug in the given values for the speed and time to calculate the distance from City A to City B.
DistanceAB=55mphtimes1.8hoursDistance_{AB} =55 \mathrm{mph} \\times1.8 \mathrm{hours}

STEP 4

Calculate the distance from City A to City B.
DistanceAB=55mphtimes1.8hours=99milesDistance_{AB} =55 \mathrm{mph} \\times1.8 \mathrm{hours} =99 \mathrm{miles}

STEP 5

Next, we need to find the distance between City B and City C. We can do this by multiplying the speed of the automobile by the time taken to drive from City B to City C.
DistanceBC=SpeedBCtimesTimeBCDistance_{BC} = Speed_{BC} \\times Time_{BC}

STEP 6

Now, plug in the given values for the speed and time to calculate the distance from City B to City C.
DistanceBC=55mphtimes1.6hoursDistance_{BC} =55 \mathrm{mph} \\times1.6 \mathrm{hours}

STEP 7

Calculate the distance from City B to City C.
DistanceBC=55mphtimes1.6hours=88milesDistance_{BC} =55 \mathrm{mph} \\times1.6 \mathrm{hours} =88 \mathrm{miles}

STEP 8

Now, we need to calculate the angle between the paths AB and BC. Since the bearing from City A to City B is 48∘48^{\circ} \mathrm{} and from City B to City C is 42∘42^{\circ} \mathrm{}, the angle between the paths AB and BC is 180∘−(48∘+42∘)180^{\circ} - (48^{\circ} +42^{\circ}).
θ=180∘−(48∘+42∘)\theta =180^{\circ} - (48^{\circ} +42^{\circ})

STEP 9

Calculate the angle between the paths AB and BC.
θ=180∘−(48∘+42∘)=90∘\theta =180^{\circ} - (48^{\circ} +42^{\circ}) =90^{\circ}

STEP 10

Now, we can use the law of cosines to find the distance from City A to City C.
DistanceAC=DistanceAB2+DistanceBC2−2×DistanceAB×DistanceBC×cos⁡(θ)Distance_{AC} = \sqrt{Distance_{AB}^2 + Distance_{BC}^2 -2 \times Distance_{AB} \times Distance_{BC} \times \cos(\theta)}

STEP 11

Plug in the values for the distances and the angle to calculate the distance from City A to City C.
DistanceAC=99+88−×99×88×cos⁡(90∘)Distance_{AC} = \sqrt{99^ +88^ - \times99 \times88 \times \cos(90^{\circ})}

STEP 12

Calculate the distance from City A to City C.
DistanceAC=992+882−2×99×88×0=992+882Distance_{AC} = \sqrt{99^2 +88^2 -2 \times99 \times88 \times0} = \sqrt{99^2 +88^2}

STEP 13

Calculate the distance from City A to City C.
DistanceAC=992+882=980+774=17545≈132milesDistance_{AC} = \sqrt{99^2 +88^2} = \sqrt{980 +774} = \sqrt{17545} \approx132 \mathrm{miles}The distance from City A to City C is approximately132 miles.

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