Question

Patty is 50 mi. due north of Todd. Emma is due east of Todd. The angle between the direction of the shortest distance from Emma and Patty and due east is 40°. To the nearest mile, how far is Emma from each person?

Answer 1

Hence, the distance between Todd and Emma is:41.955 mi.

and the distance between Patty and Emma is:32.14 mi.

Explanation:

We will see that we will get a right angled triangle based on our information(The graph is attached to the answer)

Now let x denotes the distance between Emma and Todd.

and y denotes the distance between Emma and Patty.

Now as the triangle is right angled so we can use the trignometric identities to find the distance.

`\tan40\degree=(50)/(x)`

this means
`x=\tan40\degree * 50`

Now as
`\tan40\degree=0.8391`

Hence,
`x=41.955 \text{mi.}`

Similarly,

`\sin40\degree=(50)/(y)`

Hence
`y=32.14 \text{mi.}`

Answer 2

For this it would be helpful if you draw a diagram of the situation. You will see that a right triangle can be formed from the problem. We use trigonometric functions to solve. We do as follows:

sin 40 = 50/y
tan 40 = 50/x


Solving we obtain,
x = 60 mi
y = 78 mi

Hope this answers the question. Have a nice day.