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 \(\displaystyle t = 3.\)