Harry Potter-themed Related Rates question!

[thejuansoto]

Aaa

Aaaa

 

[tkhunny]

Sounds like a good problem. What have you tried? Did you draw a picture and make some right triangles?

 

[soroban]

Hello, thejuansoto!

Harry has accidentally turned Aunt Marge into a balloon.
Aunt Marge is rising vertically above a straight road at a constant rate of 1 m/sec.
Just when she is 65 metres above the ground, a bicycle moving at a constant rate of 17 m/sec passes under her.
How fast is the distance between Aunt Marge and the bicycle increasing 3 seconds later?

    A *
      | *
    t |   *
      |     *
    P *       *  x 
      |         *
      |           *
   65 |             *
      |               *
      |                 *
    C * - - - - - - - - - * B
              17t

When Aunt Marge is at $\displaystyle P$, the bicycle pass through $\displaystyle C.$

In the next $\displaystyle t$ seconds, Aunt Marge rises $\displaystyle t$ meters to $\displaystyle A.$

In the same $\displaystyle t$ seconds, the bicycle moves $\displaystyle 17t$ meters to $\displaystyle B.$

Let $\displaystyle x \,=\,AB$

We have: .\(\displaystyle x^2 \:=\t+65)^2 + (17t)^2 \:=\:290t^2 + 130t + 4225\)


Now find $\displaystyle \dfrac{dx}{dt}$ when $\displaystyle t = 3.$