The behavior of the spring can be described using either a sine or cosine function. The spring's maximum displacement is 6 inches, occurring at t=0, which we will define as the positive peak. Therefore, we can express the function as:
6sin(at+B). The spring's period is 4 minutes, which means the time factor in the equation must complete a cycle (2π) in 4 minutes. This gives us the equation 4min*a=2π, leading to a=π/2. Generally, a=2π/T where a is the coefficient and T is the period. For B, since sin(π/2)=1, we determine B=π/2 because at t=0, the equation becomes 6sin(B)=6. Therefore, we substitute to form f(t)=6sin(πt/2+π/2)=6cos(πt/2)
due to trigonometric relations.
(5r - 4)(r² - 6r + 4)
uses the distributive property for multiplication.
This expands to 5r(r² - 6r + 4) - 4(r² - 6r + 4)
which results in 5r³ - 30r² + 20r - 4r² + 24r - 16
as you combine like terms and simplify.
The outcome is 5r³ - 30r² - 4r² + 20r + 24r - 16
leading to a final expression of 5r³ - 34r² + 44r - 16. Choice A.
C. No, because at least one of the sample sizes exceeds 10 percent of the population.
