A train first travels east for 20 kilometers. Then it turns north and travels for 43 kilometers. Next it goes west for 20 kilometers. Finally, it travels south for an unknown number of kilometers. At the end of the journey, the train is back at the original location. How far did it travel southward?

the train traveled 43km southwards

Step-by-step explanation:

First, let's define.

North as the positive y-axis.

East as the positive x-axis.

And the initial position of the train is (0, 0).

Now, we know that:

"A train first travels east for 20 kilometers."

The new position will be:

(0, 0) + (20km, 0) = (20km , 0)

"Then it turns north and travels for 43 kilometers."

The new position will be:

(20km, 0) + (0, 43km) = (20km, 43km)

"Next it goes west for 20 kilometers."

If East if the positive x-axis, then West is the negative x-axis.

The new position of the train will be:

(20km, 43km) + (-20km, 0) = (0, 43km)

Then it travels south for an unknown number of kilometers, such that the final position is the same as the initial position, then if the train goes X kilometers to the south (and because North is the positive y-axis, South will be the negative y-axis), we have:

(0, 43km) + (0, -X) = (0, 0)

(0, 43km - X) = (0, 0)

Then we must have that:

43km - X = 0

43km = X

This means that the train traveled 43km southwards.

the distance between point a and point b is 6 units

walter needs to make at least 5 more cookies but no more than 40. 5 ≤ x ≤ 40

step-by-step explanation:

20 ≤ x + 15 ≤ 55

subtract 15 from all parts

20-15 ≤ x + 15-15 ≤ 55-15

5 ≤ x ≤ 40

this means how many more cookies he has to make

walter needs to make at least 5 more cookies, but no more than 40 cookies; 5 ≤ x ≤ 40