Question

Sophie is riding her bike home when she runs over a nail. It gets stuck to the tire of her bike but does not pop the tire. As she continues to cycle home the nail hits the ground every 2 seconds and reaches a maximum height of 48cm. a) Write a sinusoidal equation that models the nails height off the ground in cm, h, in terms of time,t. Sketch one full revolution of the nail, assuming that sophie first runs over the nail at 0seconds . b) Algebraically determine the height of the nail above the ground at 0.8 seconds. Round your answer to the nearest tenth of a cm

Answer 1

a) The sinusoidal equation is;

The function is h = -24·cos[π(t )] + 24

The sketch of one full revolution is attached

b) The height of the nail at 0.8 seconds is 43.42 cm

Explanation:

The sinusoidal equation that models the nails height can be given as follows;

y = A·cos[B(x - C)]+D

A = The amplitude = Half maximum height = 48/2 = 24 cm

The period = 2·π/B = Time to complete one oscillation = 2 seconds

∴ B = 2·π/2 = π

x = t = Time

C = The horizontal shift

D = the vertical shift = 24 cm

y = The height of the nail = h

We have;

h = -24·cos[π(t - C)] + 24

At t = 0, h = 0, therefore, we have;

0 = -24·cos[π(0 - C)] + 24

24·cos[π(0 - C)] = 24

∴ cos[π(0 - C)] = 24/24 = 1

π(0 - C) = 0

C = 0

The function is h = -24·cos[π(t )] + 24

b) The height of the nail at 0.8 seconds is given as follows;

h = -24×cos[π(0.8)] + 24 =

h = 19.42 + 24 = 43.42 cm.