Question 1

Find an equation of the circle with the given center and radius.&#10;Center (1

Accepted Answer

The standard form of the equation of a circle with center $(h, k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$. 
Thus, applying this to the given center and radius, we get $(x-1)^2 + (y+3)^2 = 1^2$. 
Simplifying this, we get $(x-1)^2 + (y+3)^2 = 1$, which matches choice D.

Question 2

Solve.&#10;

Accepted Answer

This equation represents a circle. To find its center and radius, we complete the squares for x and y. The equation can be rewritten as $(x^2 - 10x) + (y^2 - 8y) + 16 = 0$. Completing the square for each variable gives $(x - 5)^2 + (y - 4)^2 = 25$, which means the center of the circle is at $(5, 4)$ and the radius $r = \sqrt{25} = 5$.

Question 3

Solve the system by the substitution method.&#10;

Accepted Answer

Substitute $y=x^2-2$ into the first equation,
$$5x^2-(x^2-2)^2=10$$
$$\Rightarrow 5x^2-x^4+4x^2-4=10$$
$$\Rightarrow x^4-9x^2+14=0$$
$$(x^2-7)(x^2-2)=0$$
Therefore, $x=\pm\sqrt{2},\pm\sqrt{7}$. Substituting these values back into $y=x^2-2$, we can find the corresponding solutions to the system of equations.
\begin{align*} &(x,y)=\left(-\sqrt{2},0ight) &\Rightarrow &\quad( - \sqrt { 2 } , 0 )\ &(x,y)=(\sqrt{2},0) &\Rightarrow &\quad(\sqrt { 2 } , 0 )\ &(x,y)=\left(-\sqrt{7},5ight) &\Rightarrow &\quad( - \sqrt { 7 } , 5 )\ &(x,y)=(\sqrt{7},5) &\Rightarrow &\quad( \sqrt { 7 } , 5 ) \end{align*}
Therefore, the correct answer is (B).

Question 4

Solve the system by the substitution method.&#10;

Accepted Answer

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Question 5

Solve the system by the substitution method.&#10;

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Question 6

Solve the system by the substitution method.&#10;

Accepted Answer

From the second equation, we can substitute y in the first equation and get an equation in terms of x only: 
$x^2 + (x^2-6)^2 = 36$
Simplifying this, we get 
$2x^4 - 12x^2 + 27 = 0$
Factoring out a 2, we get 
$2(x^2-3)^2 + 3 = 0$
But since $(x^2-3)^2$ is non-negative, this equation has no real solutions. Therefore, the system has no real solutions. However, if we were to consider complex solutions, the equation $2(x^2-3)^2 + 3 = 0$ has complex solutions $x=\pm\sqrt{3\pm i\sqrt{2}}$. Substituting these into $y=x^2-6$ gives us the complex solutions: $(\pm\sqrt{3\pm i\sqrt{2}}, 3\mp 2i\sqrt{3\pm i\sqrt{2}})$. The three real solutions in the answer choices are all incorrect.

Question 7

Solve the system by the substitution method.&#10;

Accepted Answer

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Question 8

Solve the system by the substitution method.&#10;

Accepted Answer

To solve the system by substitution, set the two equations equal to each other since they both equal $y$: $$3x^2 + 6x = x^2 - 4x + 12.$$ Rearrange the equation to set it to 0: $$2x^2 + 10x - 12 = 0.$$ Simplify by dividing the entire equation by 2: $$x^2 + 5x - 6 = 0.$$ Factor the quadratic equation: $$(x + 6)(x - 1) = 0.$$ Solve for $x$: $x = -6$ and $x = 1.$ Substitute $x = -6$ and $x = 1$ back into one of the original equations to find $y$: For $x = 1$, $y = 3(1)^2 + 6(1) = 9.$ For $x = -6$, $y = 3(-6)^2 + 6(-6) = 72.$ Therefore, the solutions are $(1, 9)$ and $(-6, 72)$.

Question 9

Solve.&#10;

Accepted Answer

The given equation can be rewritten in the form of a circle equation $(x - h)^2 + (y - k)^2 = r^2$ by completing the square for both $x$ and $y$. Completing the square for $x^2 - 8x$ gives $(x - 4)^2 - 16$, and for $y^2 - 8y$ gives $(y - 4)^2 - 16$. Adding these and equating to the original equation, we get $(x - 4)^2 + (y - 4)^2 = 16$, which represents a circle with center $(4, 4)$ and radius $r = 4$. However, there was a mistake in my calculation of the radius. The correct radius is $r = \sqrt{16} = 4$, not $2$. Therefore, the correct answer should have been identified as option C, not A, based on the correct calculation of the radius.

Question 10

Solve.&#10;

Accepted Answer

The answer of Solve.&#10; ...

Question 11

Solve the system by the substitution method.&#10;

Accepted Answer

The answer of Solve the system by the substitution method.&#10;...

Question 12

Find an equation of the circle with the given center and radius.&#10;Center (0, 0), radius 19

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The answer of Find an equation of the circle with...

Question 13

Solve the system by the substitution method.&#10;

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Question 14

Find an equation of the circle with the given center and radius.&#10;Center (0, -10), radius 5

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Question 15

Find an equation of the circle with the given center and radius.&#10;Center (-4

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Question 16

Solve.&#10; &#10;A) 10 yd by 14 yd&#10;B) 9 yd by 14 yd&#10;C) 9 yd by 13 yd&#10;D) 10 yd by 13 yd

Accepted Answer

The answer of Solve.&#10; &#10;A) 10 yd by 14 yd&#10;B)...

Question 17

Solve.&#10;

Accepted Answer

The answer of Solve.&#10; ...

Question 18

Solve.&#10; &#10;A) 16 ft by 15 ft&#10;B) 120 ft by 120 ft&#10;C) 24 ft by 10 ft&#10;D) 20 ft by 12 ft

Accepted Answer

The answer of Solve.&#10; &#10;A) 16 ft by 15 ft&#10;B)...

Question 19

Solve.&#10;

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Question 20

Solve.&#10;

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Question 21

Solve the problem.&#10;An experimental model for a suspension bridge is built. In one section, cable runs from the top of one tower down to the roadway, just touching it there, and up again to the top of a second tower.&#10;The towers are both 4 inches tall and stand 40 inches apart. At some point along the road from the&#10;Lowest point of the cable, the cable is 0.36 inches above the roadway. Find the distance between&#10;That point and the base of the nearest tower.&#10;A) 14.2 in.&#10;B) 14 in.&#10;C) 6.2 in.&#10;D) 5.8 in.

Accepted Answer

The answer of Solve the problem.&#10;An experimental model for a...

Question 22

Graph the hyperbola. Give the coordinates of the center as well as the values of a and b.
A Ferris wheel has a diameter of 320 feet and the bottom of the Ferris wheel is 11 feet above the ground. Find the equation of the wheel if the origin is placed on the ground directly below the
Center of the wheel, as illustrated.

Accepted Answer

The answer of Graph the hyperbola. Give the coordinates of...

Question 23

Find the distance between the pair of points. Round to the nearest tenth if necessary.&#10;

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Question 24

Find the distance between the pair of points. Round to the nearest tenth if necessary.&#10;

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Question 25

Graph the ellipse. Give the coordinates of the center, as well as the values of a and b.&#10;

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The answer of Graph the ellipse. Give the coordinates of...

Question 26

Find an equation of the circle with the given center and radius.&#10;

Accepted Answer

The answer of Find an equation of the circle with...

Question 27

Graph the hyperbola. Give the coordinates of the center as well as the values of a and b.
A Ferris wheel has a diameter of 320 feet and the bottom of the Ferris wheel is 11 feet above the ground. Find the equation of the wheel if the origin is placed on the ground directly below the
Center of the wheel, as illustrated.

Accepted Answer

The answer of Graph the hyperbola. Give the coordinates of...

Question 28

Find an equation of the circle with the given center and radius.&#10;Center (0, -4), radius 15

Accepted Answer

The answer of Find an equation of the circle with...

Question 29

Find the distance between the pair of points. Round to the nearest tenth if necessary.&#10;

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The answer of Find the distance between the pair of...

Question 30

Find an equation of the circle with the given center and radius.&#10;

Accepted Answer

The answer of Find an equation of the circle with...

Question 31

Graph the ellipse. Give the coordinates of the center, as well as the values of a and b.&#10;

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The answer of Graph the ellipse. Give the coordinates of...

Question 32

Find the midpoint of the line segment that connects the given points.&#10;

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Question 33

Find the distance between the pair of points. Round to the nearest tenth if necessary.&#10;

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Question 34

Find an equation of the circle with the given center and radius.&#10;

Accepted Answer

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Question 35

Find an equation of the circle with the given center and radius.&#10;

Accepted Answer

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Question 36

Find an equation of the circle with the given center and radius.&#10;

Accepted Answer

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Question 37

Find the distance between the pair of points. Round to the nearest tenth if necessary.&#10;

Accepted Answer

The answer of Find the distance between the pair of...

Question 38

Solve the problem.&#10;Arches in the shapes of parabolas are often used in construction. Find the equation of a parabolic arc that is 18 feet high at its highest point and 26 feet wide at the base, as illustrated in the figure.&#10;Place the origin at the midpoint of the base.

Accepted Answer

The answer of Solve the problem.&#10;Arches in the shapes of...

Question 39

Solve the problem.&#10;A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 154 feet and a maximum height of 30 feet. Find the height of the arch at 20 feet from its center.&#10;A) 34.2 ft&#10;B) 0.5 ft&#10;C) 28 ft&#10;D) 8.1 ft