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Questions 1 - 10

Scott wants to invest $1000 for 1 year. At Bank A, his investment will collect 3% interest compounded daily while at Bank B, his investment will collect 3.50% interest compounded monthly. Which bank offers a better return? How much more will he receive by choosing that bank over the other?

$Bank\ B;\ \$5.12$

$Bank\ A;\ \$5.12$

$Bank\ B;\ \$1005.12$

$Bank\ A;\ \$1005.12$

Explanation

Calculate the total amount from each bank using the following formula:

$A=P\left(1+\frac{r}{n}\right)^{nt}$

Bank A:

$A=1000\left(1+\frac{0.03}{365}\right)^{(1)(365)}=1030.45$

Bank B:

$A=1000\left(1+\frac{.035}{12}\right)^{(1)(12)}=1035.57$

What is the measure of one exterior angle of a regular twenty-four sided polygon?

$15^{\circ }$

$18^{\circ }$

$20^{\circ }$

$24^{\circ }$

$25^{\circ }$

Explanation

The sum of the measures of the exterior angles of any polygon, one at each vertex, is $360^{\circ }$ . Since a regular polygon with twenty-four sides has twenty-four congruent angles, and therefore, congruent exterior angles, just divide:

$360 \div 24 = 15$

Define a function as follows:

$f(x) = 2 ^{x-1} + 7$

Give the horizontal aysmptote of the graph of .

$y = -\frac{7}{2}$

Explanation

The horizontal asymptote of an exponential function can be found by noting that a positive number raised to any power must be positive. Therefore, $2 ^{x-1} > 0$ and $y= 2 ^{x-1} + 7 > 7$ for all real values of . The graph will never crosst the line of the equatin , so this is the horizontal asymptote.

What is the area of a circle with a diameter of ?

$324\pi cm^2$

$81\pi cm^2$

$54\pi cm^2$

$36\pi cm^2$

$90\pi cm^2$

Explanation

The area of a circle is defined by $A=\pi r^{2}$ , where is the radius of the circle. We are provided with the diameter of the circle, which is twice the length of .

If , then $r=\frac{d}{2}=\frac{18cm}{2}=9cm$

Then, solving for :

$A=\pi r^{2}$

$A=\pi (9cm)^{2}$

$A=81\pi cm^2$

A $60^{\circ }$ arc of a circle measures . Give the radius of this circle.

$\frac{3}{\pi}x+\frac{12}{\pi}$

$\frac{6}{\pi}x+\frac{24}{\pi}$

$6 \pi x+24 \pi$

$9 \pi x^{2}+ 72 \pi x+ 144 \pi$

$3\pi x+12\pi$

Explanation

A $60^{\circ }$ arc of a circle is $\frac{60}{360 } = \frac{1}{6}$ of the circle. Since the length of this arc is , the circumference is $\textup{six times }$ this, or

The radius of a circle is its circumference divided by $2 \pi$ ; therefore, the radius is

$r = \frac{C}{2 \pi} = \frac{6x+24}{2\pi} = \frac{3}{\pi}x+\frac{12}{\pi}$

Two circles are constructed; one is inscribed inside a given square, and the other is circumscribed about the same square.

The circumscribed circle has circumference $20 \pi$ . Give the area of the inscribed circle.

$50 \pi$

$200 \pi$

$300 \pi$

$\frac{100 \pi}{3}$

The correct answer is not among the other responses.

Explanation

Examine the diagram below, which shows the square, segments from its center to a vertex and the midpoint of a side, and the two circles.

Thingy

Note that the segment from the center of the square to the midpoint of a side is a radius of the inscribed circle, and the segment from the center to a vertex is a radius of the circumscribed circle. The two radii and half a side of the square form a 45-45-90 Triangle, so by the 45-45-90 Theorem, the radius of the inscribed circle is equal to that of the circumscribed circle divided by $\sqrt{2}$ .

The inscribed circle has circumference $20 \pi$ , so its radius is

$20 \pi \div 2 \pi = 10$

Divide this by $\sqrt{2}$ to get the radius of the circumscribed circle:

$\frac{10}{\sqrt{2}}= \frac{10 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{10 \sqrt{2}}{2} = 5 \sqrt{2}$

The circumscribed circle has area

$A= \pi r^{2} = \pi \cdot \left (5 \sqrt{2} \right ) ^{2}= \pi \cdot 25 \cdot 2 = 50 \pi$

A given circle has an area of $81\pi$ . What is the length of its diameter?

Not enough information provided

Explanation

The area of a circle is defined by the equation $A=\pi r^{2}$ , where is the length of the circle's radius. The radius, in turn, is defined by the equation $r=\frac{d}{2}$ , where is the length of the circle's diameter.

Given $A=81\pi$ , we can deduce that $r^{2}=81$ and therefore . Then, since $r=\frac{d}{2}$ , .

What percentage of a circle is a sector if the angle of the sector is $27^{\circ}$ ?

Explanation

The full measure of a circle is $360^{\circ}$ , so any sector will cover whatever fraction of the circle that its angle is of $360^{\circ}$ . We are given a sector with an angle of $27^{\circ}$ , so this sector will cover a percentage of the circle equal to whatever fraction $27^{\circ}$ is of $360^{\circ}$ . This gives us:

$\frac{27^{\circ}}{360^{\circ}}=\frac{3}{40}=0.075=7.5%$

The chord of a $60^{\circ }$ central angle of a circle with area $40 \pi$ has what length?

$2 \sqrt{10}$

$10\sqrt{2}$

$4 \sqrt{5}$

Explanation

The radius of a circle with area $40 \pi$ can be found as follows:

$\pi r^{2} = A$

$\pi r^{2} = 40 \pi$

$r^{2} = 40$

$r = \sqrt{40 } = \sqrt{4} \cdot \sqrt{10} = 2 \sqrt{10}$

The circle, the central angle, and the chord are shown below:

Chord

By way of the Isosceles Triangle Theorem, $\Delta AOB$ can be proved equilateral, so $AB = AO= 2 \sqrt{10}$ , the correct response.

On January 1, Gary borrows $10,000 to purchase an automobile at 12% annual interest, compounded quarterly beginning on April 1. He agrees to pay $800 per month on the last day of the month, beginning on January 31, over twelve months; his thirteenth payment, on the following January 31, will be the unpaid balance. How much will that thirteenth payment be?

$\$913.16$

$\$1,020.79$

$\$661.07$

$\$1,122.25$

$\$448.00$

Explanation

12% annual interest compounded quarterly is, effectively, 3% interest per quarter.

Over the course of one quarter, Gary pays off $$800 \times 3 = $2,400$ , and the remainder of the loan accruses 3% interest. This happens four times, so we will subtract $2,400 and subsequently multiply by 1.03 (adding 3% interest) four times.