Q: The location of a point is given by polar coordinates $$(-2, \frac{3π}{4})$$. Which of the following gives another representation of this point?

A negative radius can be rewritten.$$(-r,\theta)=(r,\theta+\pi)$$So,$$(-2,\frac{3\pi}{4})=(2,\frac{3\pi}{4}+\pi)$$$$=(2,\frac{7\pi}{4})$$Final answer: $$(2,\frac{7\pi}{4})$$

Q: Which of the following is equivalent to the expression $$4\left(\cos^2\left(\frac{3\pi}{5}\right)-\sin^2\left(\frac{3\pi}{5}\right)\right)$$?

Use the identity$$\cos^2\theta-\sin^2\theta=\cos(2\theta)$$So$$4\left(\cos^2\left(\frac{3\pi}{5}\right)-\sin^2\left(\frac{3\pi}{5}\right)\right)=4\cos\left(2\cdot\frac{3\pi}{5}\right)$$Simplify.$$4\cos\left(\frac{6\pi}{5}\right)$$Final answer: $$4\cos\left(\frac{6\pi}{5}\right)$$

Q: Consider a circle centered at the origin in the $$xy$$-plane. An angle of measure $$\frac{\pi}{3}$$ radians in standard position subtends an arc of length $$30$$ units. What is the radius of the circle?

Use the arc length formula.$$s=r\theta$$Substitute the given values.$$30=r\cdot\frac{\pi}{3}$$Solve for $$r$$.$$r=\frac{30}{\frac{\pi}{3}}$$$$r=30\cdot\frac{3}{\pi}$$$$r=\frac{90}{\pi}$$Final answer: $$\frac{90}{\pi}$$

Question 1

The function $$H$$ models a periodic phenomenon. The maximum value of $$H$$ is $$10$$, which occurs at $$t=0$$. The period of $$H$$ is $$8$$ hours. For $$0 ≤ t ≤ 16$$, which of the following could be an expression for $$H(t)$$?

Accepted Answer

Maximum value is $$10$$.So, amplitude plus midline equals $$10$$.Amplitude is $$5$$ and midline is $$5$$.So, the function is of the form $$5\cos(\cdot)+5$$Maximum occurs at $$t=0$$.So, cosine is used. Period is $$8$$.Use formula $$\frac{2\pi}{b}=8$$$$b=\frac{\pi}{4}$$So, the function is $$5\cos\left(\frac{\pi}{4}t\right)+5$$Final answer: $$5\cos\left(\frac{\pi}{4}t\right)+5$$

Question 2

A data set is modeled by $$k(t)=12·3^t$$. The data are plotted on a semi-log graph where the vertical axis is scaled using the natural logarithm. Which of the following describes the appearance of the data?

Accepted Answer

Take natural logarithm.$$\ln k(t)=\ln(12\cdot3^t)$$$$\ln k(t)=\ln 12 + t\ln 3$$This is linear in $$t$$.Slope is $$\ln 3$$.Final answer: Linear with slope ln 3

Question 3

The location of a point is given by polar coordinates $$(-2, \frac{3π}{4})$$. Which of the following gives another representation of this point?

Accepted Answer

A negative radius can be rewritten.$$(-r,	heta)=(r,	heta+\pi)$$So,$$(-2,\frac{3\pi}{4})=(2,\frac{3\pi}{4}+\pi)$$$$=(2,\frac{7\pi}{4})$$Final answer: $$(2,\frac{7\pi}{4})$$

Question 4

The table shows values of a function $$f$$. Which function type best models the data?

Accepted Answer

Look at the output values.$$4,9,16,25,36,49$$These are perfect squares.$$4=2^2$$ $$9=3^2$$$$16=4^2$$$$25=5^2$$$$36=6^2$$$$49=7^2$$So, the pattern is $$f(x)=(x+1)^2$$This is a quadratic function.Final answer: Quadratic

Question 5

The height of a buoy in the ocean is modeled by a sinusoidal function $$h(t)$$. The buoy reaches a maximum height of $$4$$ feet and a minimum height of $$-4$$ feet. The time between consecutive maximum heights is $$6$$ seconds. Which of the following could represent $$h(t)$$?

Accepted Answer

The amplitude is:$$\frac{4-(-4)}{2}=4$$The midline is:$$\frac{4+(-4)}{2}=0$$So, there is no vertical shift.The period is the time between consecutive maximum heights.$$T=6$$Use:$$\frac{2\pi}{b}=6$$Solve for $$b$$:$$b=\frac{2\pi}{6}$$$$b=\frac{\pi}{3}$$So, a matching function is:$$4\sin\left(\frac{\pi}{3}t\right)$$Final answer: $$4\sin\left(\frac{\pi}{3}t\right)$$

Question 6

Which of the following is equivalent to the expression $$4\left(\cos^2\left(\frac{3\pi}{5}\right)-\sin^2\left(\frac{3\pi}{5}\right)\right)$$?

Accepted Answer

Use the identity$$\cos^2	heta-\sin^2	heta=\cos(2	heta)$$So$$4\left(\cos^2\left(\frac{3\pi}{5}ight)-\sin^2\left(\frac{3\pi}{5}ight)ight)=4\cos\left(2\cdot\frac{3\pi}{5}ight)$$Simplify.$$4\cos\left(\frac{6\pi}{5}ight)$$Final answer: $$4\cos\left(\frac{6\pi}{5}ight)$$

Question 7

The population of a town is decreasing over time. The population is modeled by the exponential function $$P(t)=15000(0.85)^t$$, where $$t$$ is measured in years since 2025. A revised model is given by $$Q(t)=15000(1.1)(0.85)^t$$. Which of the following describes the relationship between model $$Q$$ and model $$P$$?

Accepted Answer

Write both models.$$P(t)=15000(0.85)^t$$ $$Q(t)=15000(1.1)(0.85)^t$$So$$Q(t)=1.1\cdot P(t)$$The growth factor remains $$0.85$$The coefficient in front is multiplied by $$1.1$$So, the initial value is increased by a factor of $$1.1$$Final answer: The initial value is increased by a factor of 1.1.

Question 8

Consider a circle centered at the origin in the $$xy$$-plane. An angle of measure $$\frac{\pi}{3}$$ radians in standard position subtends an arc of length $$30$$ units. What is the radius of the circle?

Accepted Answer

Use the arc length formula.$$s=r	heta$$Substitute the given values.$$30=r\cdot\frac{\pi}{3}$$Solve for $$r$$.$$r=\frac{30}{\frac{\pi}{3}}$$$$r=30\cdot\frac{3}{\pi}$$$$r=\frac{90}{\pi}$$Final answer: $$\frac{90}{\pi}$$

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