Calculus

posted by .

A piece of elastic is attached to two nails on a flat board, with a button attached to the midpoint of the elastic. The nails are 5 cm apart. You stretch the elastic by pulling the button along the board in a direction that is perpendicular to the line between the nails.
A. Fnd an equation that relates the total length of the elastic x to the distance y that the button has moved.
B. You pull the button at a constant 3 cm/sec. Find the rate at which the length of the elastic is increasing when it is 12 cm long.

  • Calculus -

    you have two right triangles of height y

    2.5^2 + y^2 = (x/2)^2
    or
    25 + 4y^2 = x^2

    8y dy/dt = 2x dx/dt
    when x = 12, y = 5.454

    8(5.454) = 2(12) dx/dt
    dx/dt = 1.818

  • Calculus -

    First, let's sketch what we can derive geometrically.
    |dw:1355290130364:dw|
    (A) Given we know those two triangles are right, we can relate the hypotenuse \(\frac12x\) to the altitude \(y\) and base \(\frac52\) using the Pythagorean theorem, which we can rearrange to yield an adequate relation:$$\left(\frac12x\right)^2=y^2+\left(\frac52\right)^2\\\frac14x^2=y^2+\frac{25}4\\x^2=4y^2+25$$
    (B) We're given that the button is moving at a rate of 3 cm/s, which can be expressed using a time derivative as \(\frac{dy}{dt}=3\). We're told that the elastic (at the instant we're interested in) is 12 cm long, i.e. \(x=12\); given this, we can determine the distance of the button from its initial position with relative ease:$$(12)^2=4y^2+25\\4y^2=144-25=119\\y^2=\frac{119}{4}\\y=\frac{\sqrt{119}}2$$Let's use implicit differentiation on our formula above to relate the rates of elongation:$$2x\frac{dx}{dt}=8y\frac{dy}{dt}\\24\frac{dx}{dt}=12\sqrt{119}\\2\frac{dx}{dt}=\sqrt{119}\\\frac{dx}{dt}=\frac{\sqrt{119}}2\approx5.4544$$

  • Calculus -

    each side of the elastic is the hypotenuse of a triangle with legs 5/2 and y, so

    x = 2√(2.5^2 + y^2)

    dx/dt = 2y/√(2.5^2+y^2) dy/dt
    when x=12, y=5.45, so
    dx/dt = 2(5.45)/6 (-3) = -1/5.45 = -0.18 cm/s

  • Calculus -

    A.
    x = 2√[ y^2 + (2.5)^2 ] cm (using Pythagorus theorem)

    B.
    x^2 = 4y^2 + 25
    => 2x dx/dt = 8y dy/dt
    => dx/dt = 4 (y/x) dy/dt
    When x = 12 cm, y = (1/2)√[ 144 - 25 ] = 2.727 cm
    => dx/dt = 4 (2.727/12) x 3 cm/s
    = 2.727 cm/s.

  • Calculus -

    A.
    x = 2√[ y^2 + (2.5)^2 ] cm (using Pythagorus theorem)

    B.
    x^2 = 4y^2 + 25
    => 2x dx/dt = 8y dy/dt
    => dx/dt = 4 (y/x) dy/dt
    When x = 12 cm, y = (1/2)√[ 144 - 25 ] = 2.727 cm
    => dx/dt = 4 (2.727/12) x 3 cm/s
    = 2.727 cm/s.

  • Calculus -

    Pythagorous Theorem,

    z^2 = y^2 + x^2, where y = 2.5 cm and x is the legth of the pull.

    substituting z^2 = 2.5^2 +x^2, z =sqrt(x^2 +6.25)

    dz/dx (2z) = 2x

    dz/dx = (x/z) ......... (a)

    but dz/dt = dz/dx * dx/dt

    Thus, dz/dt = (x/z) * dx/dt = x/z * 3

    when z = 12 , y = 5, x= sqrt(144-25) = sqrt(119)

    Thus dz/dt = sqrt(119)/12 * 3
    = sqrt(119)/4 =10.91 /4 =2.73 cm/sec

    Rate of increase in length of elastic = 2.73 cm/sec

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. physics

    A force of 50.0N is used to stretch an elastic so that it had 40.0 J of elastic potential energy. Through what distance was the elastic stretched?
  2. Physics

    A 0.25-kg ball is attached to a 26-cm piece of string. The ball is first raised so that the string is taut and horizontal, then the ball is released so that, at the bottom of its swing, it undergoes an elastic head-on collision with …
  3. economics

    Wheat farmers in Kansas would benefit from a devastating crop failure in N.Dakota if the U.S.demand for wheat?
  4. college

    A 3.2 kg block is hanging stationary from the end of a vertical spring that is attached to the ceiling and has an elastic potential energy of 1.8 J. What is the elastic potential energy of the system when the 3.2 kg block is replaced …
  5. Calculus (demand)

    18) The demand equation is x + 1/6p - 10+0. Compute the elasticity of demand and determine whether the demand is elastic, unitary, or inelastic at p=50. a) 5; elastic b) 1/9; inelastic c) 7/6; elastic d) 6; elastic I choose answer …
  6. calculus

    18) The demand equation is x + 1/6p - 10+0. Compute the elasticity of demand and determine whether the demand is elastic, unitary, or inelastic at p=50. a) 5; elastic b) 1/9; inelastic c) 7/6; elastic d) 6; elastic I choose answer …
  7. Chemistry

    Typically, there is a scale provided for weighing the nails. For example, a notice placed above the nail bin might read, “For the nails in the bin below, there are 500 nails per kg.” Using this conversion factor, perform the following …
  8. calculus

    A piece of elastic is attached to two nails on a flat board, with a button attached to the midpoint of the elastic. The ails are 5 cm apart. You stretch the elastic by pulling the button along the board i the direction that is perpendicular …
  9. chem

    Consider two identical iron nails: One nail is heated to 95 °C, the other is cooled to 15 °C. The two nails are placed in a coffee cup calorimeter and the system is allowed to come to thermal equilibrium. What is the final temperature …
  10. Math

    a light elastic string attached to two points A and B on the same horizontal level has a wall picture of mass 1.4kg attached to its midpoint. The weight of the wall picture causes the elastic card of 40cm to a new length of 50cm. Draw …

More Similar Questions