Question 1

A vertical cross section of a cooling tower is a portion of a hyperbola, as shown in the diagram below. The standard form of the equation of the hyperbolic cross section is  where x and y are measured in feet. The horizontal cross sections of the tower are circles.  Find the radius of the top of the tower. Round to the nearest foot.
A)90 feet
B)88 feet
C)93 feet
D)87 feet
E)85 feet

Accepted Answer

The answer of A vertical cross section of a cooling...

Question 2

When a plane exceeds the speed of sound, a sonic boom is produced by the wake of the sound waves. For a plane flying at 13,000 feet, the circular wave front can be given by  where z is the height of the wave front above Earth. See the diagram below. Note that the xy -plane is Earth s surface, which is approximately flat over small distances. Find and name the equation formed when the wave front hits Earth. 
A) 
B)  e
C) 
D) 
E)

Accepted Answer

When the wave front hits Earth, it will be at z=0. Therefore, we plug in z=0 into the given equation and get:
x^2 + y^2 = (132,000)^2
which is the equation of a circle with radius 132,000 feet centered at the origin. The correct choice is D.

Question 3

Identify the graph of the equation as a parabola, an ellipse, or a hyperbola. 
A)Ellipse
B)Hyperbola
C)Parabola

Accepted Answer

The answer of Identify the graph of the equation as...

Question 4

Plot the point on a polar coordinate system. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The given point is represented by a hexadecimal number in the format of a UUID, which suggests that it is not a polar coordinate. Therefore, any of the given options cannot be plotted on a polar coordinate system.

Question 5

Use the definition of a hyperbola to find the equation of the hyperbola in standard form. foci (7, 0)and ( - 7, 0); passes through the point 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use the definition of a hyperbola to...

Question 6

A vertical cross section of a cooling tower is a portion of a hyperbola, as shown in the diagram below. The standard form of the equation of the hyperbolic cross section is  where x and y are measured in feet. The horizontal cross sections of the tower are circles.  What is the smallest radius of a horizontal cross section? Round to the nearest foot.
A)52 feet
B)50 feet
C)72 feet
D)54 feet
E)60 feet

Accepted Answer

The answer of A vertical cross section of a cooling...

Question 7

Use the Conic Identification Theorem to identify the graph of the equation as a parabola, an ellipse, or a hyperbola. 
A)Hyperbola
B)Parabola
C)Ellipse

Accepted Answer

The equation given is in the form of (x-h)^2/a^2 - (y-k)^2/b^2 = 1, which is the standard form of a hyperbola.

Question 8

Use the eccentricity to find the equation of the hyperbola in standard form. center ( - 2, - 6), conjugate axis of length 8 with the transverse axis parallel to the x -axis, eccentricity 3
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use the eccentricity to find the equation...

Question 9

Use the definition of a hyperbola to find the equation of the hyperbola in standard form. foci (9, 0)and ( - 9, 0); passes through the point (9, 24)
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Use the definition of a hyperbola to...

Question 10

Graph the equation. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The options given are not valid equations. They seem to be a mix of hexadecimal codes and letters. As such, it is not possible to graph any of them.

Question 11

Use the Conic Identification Theorem to identify the graph of the equation as a parabola, an ellipse, or a hyperbola. 
A)Hyperbola
B)Parabola
C)Ellipse

Accepted Answer

Based on the Conic Identification Theorem, the equation with the given coefficients represents a hyperbola because the variables have opposite signs and the coefficient of the squared term is non-zero.

Question 12

Use the eccentricity to find the equation of the hyperbola in standard form. eccentricity 2, foci (12, 0)and ( - 12, 0)
A) 
B) 
C) 
D) 
E)

Accepted Answer

The standard form of the hyperbola with center at the origin is:
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
The distance between the center and each focus is equal to $c$, where $c$ is the distance between each focus and the vertex.
In this case, the center is at $(0,0)$, the foci are at $(12,0)$ and $(-12,0)$, and the eccentricity is 2. 
Since $eccentricity = \frac{c}{a}$, we have $2 = \frac{c}{a}$. 
Also, we know that $c = 12$, so $a = \frac{c}{2} = 6$. 
Using $a$ and $b$, we can write the equation of the hyperbola in standard form:
$\frac{x^2}{36}-\frac{y^2}{b^2}=1$
We only need to find $b$. The distance between each vertex and each focus is given by $ae$, where $e$ is the eccentricity. In this case, $ae = 2a = 12$. Also, the distance between each vertex and each focus is equal to $b$. So, $b=12$. 
The final equation in standard form is:
$\frac{x^2}{36}-\frac{y^2}{144}=1$
Therefore, the best choice is (C).

Question 13

Find the acute angle of rotation a that eliminates the xy term. State approximate solution to the nearest 0.1 . 
A)75.5 °
B)-33.4 °
C)14.5 °
D)22.5 °
E)-22.5 °

Accepted Answer

The answer of Find the acute angle of rotation a...

Question 14

Find the equation in standard form of a hyperbola with vertices ( 1, 1)and ( 1, 5)and eccentricity  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the equation in standard form of...

Question 15

Identify the graph of the equation below as a parabola, an ellipse, or a hyperbola. 
A)a parabola
B)an ellipse
C)a hyperbola

Accepted Answer

The equation given is typically in the form of a hyperbola. It has both x and y squared terms with different coefficients and a negative constant term. Additionally, the constants are not equal, which would mean it is not an ellipse. The absence of an x or y squared term with a positive coefficient rules out a parabola.

Question 16

Two radio transmitters are positioned along the coast, 600 miles apart as shown below in the diagram (with d equal to 600 miles). A signal is sent simultaneously from each transmitter. The signal from transmitter  is received by a ship s LORAN 700 microseconds after the ship receives the signal from  . The radio signals travel at 0.186 mile per microsecond. Find an equation of a hyperbola, with foci at  and  , on which the ship is located. Round parameters to the nearest tenth. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Two radio transmitters are positioned along the...

Question 17

Find the acute angle of rotation a that eliminates the xy term. State approximate solution to the nearest 0.1 . 
A)76.7 °
B)13.3 °
C)9.2 °
D)67.5 °
E)22.5 °

Accepted Answer

To eliminate the xy term, we need to rotate the coordinate system such that the line y=x gets aligned with one of the coordinate axes. This can be achieved by rotating the original coordinate system by an angle of 45 degrees (pi/4 radians) clockwise or counterclockwise. Since we want the acute angle of rotation, we choose the smaller angle which is 45-22.5=22.5 degrees, or pi/8 radians, counterclockwise. Therefore, the answer is D) 67.5 degrees.

Question 18

Sketch the graph of the polar equation. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Sketch the graph of the polar equation....

Question 19

Two radio transmitters are positioned along the coast, 250 miles apart. A signal is sent simultaneously from each transmitter. The signal from transmitter  is received by a ship s LORAN 300 microseconds after the ship receives the signal from  . The radio signals travel at 0.186 mile per microsecond.  If the ship is 200 miles east of the y-axis, determine its distance from the coastline (to the nearest mile).
A)865 miles
B)1000 miles
C)459 miles
D)1406 miles
E)1541 miles

Accepted Answer

The answer of Two radio transmitters are positioned along the...

Question 20

Two radio transmitters are positioned along the coast, 250 miles apart. A signal is sent simultaneously from each transmitter. The signal from transmitter  is received by a ship s LORAN 800 microseconds after the ship receives the signal from  . The radio signals travel at 0.186 mile per microsecond.  Find an equation of a hyperbola, with foci at  and  on which the ship is located. Round parameters to the nearest hundredth.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Two radio transmitters are positioned along the...

Quiz 8: Topics in Analytic Geometry

Identify the graph of the equation as a parabola, an ellipse, or a hyperbola.

Plot the point on a polar coordinate system.

Use the definition of a hyperbola to find the equation of the hyperbola in standard form. foci (7, 0) and ( - 7, 0) ; passes through the point

Use the Conic Identification Theorem to identify the graph of the equation as a parabola, an ellipse, or a hyperbola.

Use the eccentricity to find the equation of the hyperbola in standard form. center ( - 2, - 6) , conjugate axis of length 8 with the transverse axis parallel to the x -axis, eccentricity 3

Use the definition of a hyperbola to find the equation of the hyperbola in standard form. foci (9, 0) and ( - 9, 0) ; passes through the point (9, 24)

Graph the equation.

Use the Conic Identification Theorem to identify the graph of the equation as a parabola, an ellipse, or a hyperbola.

Use the eccentricity to find the equation of the hyperbola in standard form. eccentricity 2, foci (12, 0) and ( - 12, 0)

Find the acute angle of rotation a that eliminates the xy term. State approximate solution to the nearest 0.1 .

Find the equation in standard form of a hyperbola with vertices ( 1, 1) and ( 1, 5) and eccentricity .

Identify the graph of the equation below as a parabola, an ellipse, or a hyperbola.

Find the acute angle of rotation a that eliminates the xy term. State approximate solution to the nearest 0.1 .

Sketch the graph of the polar equation.