Question

The graph shows the height of a scratch on the edge of a circular gear.

Which function is the best model for the height of the scratch?
a. h(t) = 3.5 sin (π t) + 1.5
b. g(t) = 1.5 sin (π t) +3.5
c. h(t) = 1.5 sin (2 π t) + 3.5
d. h(t) = 1.5 sin (π/2 t) + 3.5

Answer 1

b. g(t) = 1.5 sin (π t) +3.5

Explanation:

You want to choose the function that has the given graph.

Test points

At t = 0, the graph shows a value of 3.5. The sine of 0 is 0, so this eliminates choice A.

At t = 1/2, the graph shows a value of 5. The values given by the different formulas are ...

b. g(1/2) = 1.5·sin(π/2) +3.5 = 5 . . . . . matches the graph

c. h(1/2) = 1.5·sin(π) + 3.5 = 3.5 . . . . no match

d. h(1/2) = 1.5·sin(π/4) +3.5 = 0.75√2 +3.5 . . . . no match

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Additional comment

The horizontal distance for one period of the graph (from peak to peak, for example) is T = 2 seconds. If the sine function is sin(ωt), then the value of ω is ...

ω = 2π/T = 2π/2 = π

This tells you the function g(t) = 1.5·sin(πt)+3.5 is the correct choice.