Question

Marissa is watching a honeybee return from her flower garden to a beehive of a branch nearby tree. She notices that the bee's path resembles part of a trigonometric function (see below). Suppose that the flower is 2 ft above the ground, the beehive is 8 ft above the ground, and the horizontal distance from the flower to the beehive is 20 ft. Model this path with a sine or cosine function. Clearly indicate the following:
a. the maximum and minimum
b. the midline
c. the period and rate constant
d. Write formula for the function
e. Clearly label all parts
f. Sketch the graph

May check your work by graphing your function on a graphing calculator etc.

Answer 1

$\bold{Answer:\quad y=-3\cos\bigg(\dfrac{\pi}{20}x\bigg)+5}$

Step-by-step explanation:

Use the formula y = A cos (Bx - C) + D    where

• A = amplitude
• Period = 2π/B
• Phase Shift = C/B
• D = vertical shift (aka midline)

Given: Max = 8, Min = 2, (1/2)Period = 20 → Period = 40

Amplitude (A) = (Max - Min)/2

                      = (8 - 2)/2

                      = 6/2

                      = 3

Midline (D) = (Max + Min)/2  

                  = (8 + 2)/2

                  = 10/2

                  = 5

Period = 2π/B    

→ B = 2π/Period

      = 2π/40

      = π/20

Notice that the Minimum touches the y-axis (not the Max) so there is no phase shift but there is a reflection → C-value = 0 & A-value is negative

Now, let's put it all together:

A = -3, B = π/20, C = 0, D = 5

$\large\boxed{y=-3\cos\bigg(\dfrac{\pi}{20}x\bigg)+5}$