Question

With a certain tail wind a jet aircraft arrives at its destination, 1,890 miles away, in 3 hours. Flying against the same wind, the plane makes the return trip in 3 3/8 hours. Find the wind speed and the plane's airspeed.

{wind speed is a0 mph, airspeed is a1 mph}

Answer 1

We assume that we can add or subtract the wind speed and the air speed directly. Relativistic effects are unlikely here :-)

Flying with the wind, the speed of the plane = air speed + wind speed
Flying against the wind, the speed of the plane = air speed - wind speed
Distance traveled is the same in both directions, 1890 miles
Time flying with the wind is 3 hours
Time flying against the wind is 3 3/8 hours

distance = velocity * time

Writing the equations with more formal notation:

\[D = 1890\]
\[D = (a_1+ a_0) * t_{with}\]\[D = (a_1 - a_0)*t_{against}\]

If we use the distributive property on both equations, we have two equations in two unknowns which can be solved by substitution or elimination.

Plug both answers into the original formula to make sure they work!

Answer 2

Wait, i don't understand. what is the actual answer for the with and against?

Answer 3

There are two answers to be found: the value of a_0 and a_1 (wind speed and air speed). t_with is the time it takes to fly the 1890 miles flying with the wind; t_against is the time it takes to fly the 1890 miles flying against the wind.

Answer 4

i know that. and that's what it's asking for.
But i dont know how to get it.

Answer 5

After you distribute, you'll have
\[D = a_1*t_{with} + a_0 * t_{with}\]\[D = a_1*t_{against} - a_0*t_{against}\]
Then you plug in the numbers and grind out the answer.

Answer 6

i don't know how to plug in the answer.

Answer 7

I would multiply the first equation by t_against/t_with so that when you add the two equations, the a_0 terms combine to 0. That will give you an equation only in a_1, which you solve for a_1. Then you plug that value into either equation and solve for a_0.

Answer 8

I'm so confused.... You have to keep in mind, I'm in high school....

Answer 9

Or do it by substitution. Maybe that's easier to understand here.
\[D = a_1*t_{with} + a_0*t_{with}\] Let's solve that to get a_0 all by itself
\[D - a_1*t_{with} = a_0*t_{with}\]
\[\frac{D - a_1*t_{with}}{a_0} = t_{with}\]
Agreed?

Answer 10

Nuts, that wasn't what I wanted, I wanted
\[\frac{D - a_1*t_{with}}{t_{with}} = a_0\]

Answer 11

I could do these problems in my sleep in high school, and with a little practice, you'll be able to as well.

Answer 12

i don't know. I've never been good at math. I can't grasp this concept of how to do it.:/

Answer 13

So, do you see how I got an expression for a_0 in terms of a_1 and our known quantities?

Answer 14

That's okay, I'm a very patient explainer :-) What part of this whole problem *do* you understand? "None" is an acceptable answer :-)

Answer 15

A man invests $5,200, part at 4% and the balance at 3%. If his total income for the two investments is $194, how much money did he invest at each rate?

$a0 at 4% and $a1 at 3%

Answer 16

and i dont understand where the a0 and a1 comes in at... I'm just completely lost when it comes to this concept.

Answer 17

Okay. Let's switch to the money problem. The man has $5200 to invest. He invests one part of it at 4% (we'll call that part a_0), and he invests the other part it at 3% (we'll call that part a_1). a_0 and a_1 are just variables, quantities that we want to find. So far, so good?

Answer 18

Hello?

Answer 19

yes

Answer 20

What do we know about a_0 and a_1? Well, we know that a_0 + a_1 = 5200, right?

Answer 21

yes

Answer 22

Please write the equation that shows the interest from the investment of a_0?

Answer 23

a_0 is invested at 4%

Answer 24

I'm not keeping you from anything important, am I?