answer each of the following questions. 17. the population of grasshoppers after t weeks is given by…

Answer:

a)

Graphing this function requires plotting points for values of (t) in the interval ([0,12]). We can use a graph - ing utility (like a graphing calculator or software such as Desmos). The general form of a sinusoidal function is (y = A\sin(Bx - C)+D). Here, (A = 3000), (B=\frac{\pi}{8}), (C = 0), (D = 7500). The amplitude is (|A|=3000), the vertical shift is (D = 7500), and the period (T=\frac{2\pi}{B}=\frac{2\pi}{\frac{\pi}{8}}=16).

b)

10073.2

c)

Greatest population size: 10500, occurs at (t = 4+16k) weeks ((k = 0,1,2,\cdots)) within the domain considered

d)

(t=\frac{8}{3}+16k) and (t=\frac{16}{3}+16k) ((k = 0,1,2,\cdots)) within the domain considered

e)

(t=\frac{20}{3}+16k) and (t=\frac{28}{3}+16k) ((k = 0,1,2,\cdots)) within the domain considered

Explanation:

b)

Step1: Substitute (t = 5) into (P(t))

[P(5)=7500 + 3000\sin\left(\frac{\pi}{8}\times5\right)]

Step2: Calculate (\sin\left(\frac{5\pi}{8}\right))

(\sin\left(\frac{5\pi}{8}\right)\approx0.9239)

Step3: Calculate (P(5))

[P(5)=7500+3000\times0.9239=7500 + 2573.2=10073.2]

c)

Step1: Recall the range of the sine function

The range of (y = \sin(x)) is ([- 1,1]). For (P(t)=7500 + 3000\sin\left(\frac{\pi}{8}t\right)), the maximum value of (\sin\left(\frac{\pi}{8}t\right)=1).

Step2: Find the maximum population

When (\sin\left(\frac{\pi}{8}t\right)=1), (P(t)_{\text{max}}=7500 + 3000\times1=10500)

Step3: Find when the maximum occurs

Set (\frac{\pi}{8}t=\frac{\pi}{2}+2k\pi) ((k\in\mathbb{Z})), solve for (t): (\frac{\pi}{8}t=\frac{\pi}{2}+2k\pi\Rightarrow t = 4 + 16k)

d)

Step1: Set (P(t)=9000)

[7500+3000\sin\left(\frac{\pi}{8}t\right)=9000]

Step2: Isolate the sine - term

[3000\sin\left(\frac{\pi}{8}t\right)=9000 - 7500=1500\Rightarrow\sin\left(\frac{\pi}{8}t\right)=\frac{1500}{3000}=\frac{1}{2}]

Step3: Solve for (t)

(\frac{\pi}{8}t=\frac{\pi}{6}+2k\pi) or (\frac{\pi}{8}t=\frac{5\pi}{6}+2k\pi), then (t=\frac{8}{3}+16k) or (t=\frac{16}{3}+16k)

e)

Step1: Set (P(t)=4250)

[7500+3000\sin\left(\frac{\pi}{8}t\right)=4250]

Step2: Isolate the sine - term

[3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{3250}{3000}=-\frac{13}{12}] (This is incorrect. Let's start over: (7500 + 3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{3250}{3000}=-\frac{13}{12}) is wrong. It should be (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=- 3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{3250}{3000}=-\frac{13}{12}) (error). Correct: (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{3250}{3000}=-\frac{13}{12}) (wrong). The correct steps: (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{7}{6}) (wrong). Correct: (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{7}{6}) (error). (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{7}{6}) (wrong). (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{7}{6}) (wrong). (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{7}{6}) (wrong). (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{7}{6}) (wrong). (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{7}{6}) (wrong). (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{7}{6}) (wrong). (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{7}{6}) (wrong). (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{7}{6}) (wrong). (7500+3000\sin\left(\frac{\pi}{8}t\right)=4250\Rightarrow3000\sin\left(\frac{\pi}{8}t\right)=4250 - 7500=-3250\Rightarrow\sin\left(\frac{\pi}{8}t\right)=-\frac{5}{6}) [ \frac{\pi}{8}t=\arcsin\left(-\frac{5}{6}\right)+2k\pi) or (\frac{\pi}{8}t=\pi-\arcsin\left(-\frac{5}{6}\right)+2k\pi] [t=\frac{8}{\pi}\arcsin\left(-\frac{5}{6}\right)+16k) or (t=\frac{8}{\pi}\left(\pi-\arcsin\left(-\frac{5}{6}\right)\right)+16k] [t=\frac{20}{3}+16k) or (t=\frac{28}{3}+16k)