A person is riding in a hot air balloon. For the first hour the wind current is

Question

A person is riding in a hot air balloon. For the first hour the wind current is a constant 9.75 miles per hour in the direction N 37.5° E. Then the wind current changes to 8.00 miles per hour and heads the balloon in the direction S 52.5° E (see the figure below).

If this continues for another 1.5 hours, how far is the balloon from its starting point? (Round your answer to one decimal place.)
___________mi

What is the bearing of the balloon from its starting point? (Round your answer to one decimal place.)
N ________° E

Explanation / Answer

Solution:- the ballon travels 9.75mph * 1h = 9.75 m

The ending coordinates are:
x1 = 9.75*cos(90-37.5) = 5.935423933
y1 = 9.75*sin(90-37.5) = 7.735195068

For the second leg, the ballon travels: 8mph * 1.5h = 12m
The *change* in coordinates are:
x2 = 12 * cos(90-52) = 9.456129048
y2 = 12 * sin(90-52) = 7.387937700

To determine the combined (final) coordinates, We probably need to look at a drawing to see
that because of the angles, the "x" coordinates combine (both heading east). However, the "y" coordinates do NOT, because the first heads north and the second heads south (so we need to subtract the second from the first).
Thus, the combined coordinates are

x = x1+x2
x = 9.75*cos(90-37.5) + 12 *cos(90-52)
x = 9.75*cos(52.5) + 12 *cos(38)
x = 9.75*0.608761429 + 12 * 0.788010754
x = 15.39155298
Also y = y1 - y2
y = 9.75*sin(90-37.5) - 12 * sin(90-52)
y = 9.75*sin(52.5) - 12 * sin(38)
y = 9.75*0.793353340 - 12 * 0.615661475
y = 0.347257364
So, the final distance is: d = sqrt(x^2 + y^2)
d = sqrt((15.39155298)^2 + (0.347257364)^2)
d = 15.39546 = 15.4

To calculate the final angle:
tan(90-angle) = y/x
tan(90-angle) = 0.347257364/15.39155298
90-angle = arctan(.02256)
90-angle = 1.29246
90 -1.29246= angle
88.7075 = angle that is angle = 88.7
where the angle is measured as "east of north"
(ie. North, 88.7 degrees east)

You may round frm starting if you want

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