I think the book is wrong?

Question

I'm tutoring a student in algebra, and one of the questions doesn't seem to work with the answer given in the answer guide.

The question is:

A person can see 72 miles to the horizon from an airplane window. How high is the airplane? Use the formula V = 3.5*SQRT(h)

The answer given is 3600 ft.

No other information should be needed (like the diameter/circumference of the earth as this is a basic college algebra class). Even trying to convert between feet and miles in different ways, I can't get the answer.

So far, you're confirming what I believed. I was afraid I was missing something simple.

Answer 1

V = 3.5√h

so h = (V/3.5)^2
= (72/3.5)^2
= 423.2

To get an answer of 3600, V = 1.2√h

Answer 2

Hi, firemedicgm.
The book's answer is right. The formula they gave you is wrong.

The correct formula is the one that Wal C found by working backwards. I arrived at that same formula by solving the problem, "How do you find the distance to the horizon (in miles), given the altitude (in feet) of an aircraft?"

What I actually found is that the formula should be V = 1.23 sqrt(h).
I really can't guess where they got the 3.5 coefficient.

Incidentally, Cirric seems to be trying to do the analysis that I did, but he's confused about which triangle to look at. To explain:
The right angle is at the horizon.
One leg is the earth's radius (from a point on the horizon to the center of the earth).
The other leg is the distance from the plane to a point on the horizon.
The hypotenuse is equal to the earth's radius plus the plane's altitude.

Plug in an accurate value for the earth's radius (3,963 miles), and you can (with a few simplifying assumptions) develop the formula shown above. (Don't forget to convert from miles to feet (for the altitude only ... everything else remains in miles).)

Answer 3

Hi. You have a right triangle with h being an unknown side. The hypotenuse is 72 miles. Square it. Your are given the second side as 3.5* square root of h. Does this help? To prove the book wrong, use the 3,600 as one side, 72 miles as the hypotenuse, and calculate the Earth' radius from the tangent point.

Answer 4

The source below explains quite clearly.

The formula is not wrong, unless there is no mention about the unit to be used in the formula.

This formula is used with V in km and h in m.

The constant 3.5 is an estimate of the expression sqrt((2R + h))/1000, by assuming h is negligible in this expression, i.e. 3.5 is an approximation for sqrt(2R), where R is the radius of the earth, when R = 6371000 m.

The division by 1000 is to convert m to km. Be aware of the division by 5280 to convert ft to miles.

When using miles and feet for V and h, R = 20902231 ft. So an estimation for sqrt(2R)/5280 would be
= sqrt(2*20902231)/5280
= 1.225

Thus, the ratio of the 2 constants is
(sqrt(2*(R in m))/1000) / sqrt(2*(R in ft))/5280)
= sqrt(2*(R in m)) / (sqrt(2 * 3.2808399 * (R in m))/5.28)
= 1 / (sqrt(3.2808399)/5.28)
= 5.28/sqrt(3.2808399)
= 2.915

Answer 5

72 = 3.5x sqrt (h)
72/3.5 = sqrt (h)

(72/3.5)^2 = h
I get 423 for h

I looked up the formula on line and it said

The formula for determining how many miles an individual can see at higher levels is the square root of his altitude (in feet) times 1.225.

So 72 = 1.225 sqrt H not 3.5

72/1.225 = sqrt H

3454 = H

Answer 6

with that formula and v =72 it should be 423.2

Answer 7

Books have errors. I would try to contact the author directly, or the publisher.