Question 1

Answer the question.
-If 1 is substituted for $x$ in the rational expression $\frac { x - 1 } { x ^ { 2 } - 1 }$, the result is $\frac { 0 } { 0 }$. Mathematicians have been known to say "Any number divided by itself is 1 ." Does this mean that this expression is equal to 1 for $x = 1$ ? Explain why or why not.

Accepted Answer

the expression is not equal to 1 for $x = 1$. When $x = 1$, the denominator becomes $1^2 - 1 = 0$, which means the expression is undefined. The result of $\frac {0} {0}$ does not have a specific value and cannot be simplified to 1. Therefore, the expression is not equal to 1 for $x = 1$.

Question 2

Answer the question.
-For $x \neq 5$, the rational expression $\frac { 8 ( x - 5 ) } { x - 5 }$ is equal to 8 . Can the same be said for $\frac { 8 x - 5 } { x - 5 }$ ? Explain why or why not.

Accepted Answer

The answer of Answer the question.
-For $x \neq 5$, the...

Question 3

Answer the question.
-Explain in your own words how to divide rational expressions. (How would you explain this to a student in this class who was absent from class?).

Accepted Answer

The answer of Answer the question.
-Explain in your own words...

Question 4

Solve.
-The volume $\mathrm { V }$ of a given mass of gas varies directly as the temperature $\mathrm { T }$ and inversely as the pressure $\mathrm { P }$. If $\mathrm { V } = 165.0 \mathrm { in } ^ { 3 }$ when $\mathrm { T } = 110 ^ { \circ }$ and $\mathrm { P } = 10 \mathrm { lb } / \mathrm { in } ^ { 2 }$, what is the volume when $\mathrm { T } = 120 ^ { \circ }$ and $\mathrm { P } = 10 \mathrm { lb } / \mathrm { in } ^ { 2 }$ ?
A) $170 \text { in. }^ 3$
B) 200 in. $^ { 3 }$
C) 160 in. $^ { 3 }$
D) $180 \text { in. }^ 3$

Accepted Answer

The answer of Solve.
-The volume $\mathrm { V }$ of...

Question 5

Solve.
-At a fixed temperature, the resistance R of a wire varies directly as the length l and inversely as the square of its diameter d. If the resistance is 0.4 ohm when the diameter is 1 mm and the length is 200 cm, what is the
Resistance when the diameter is 2 mm and the length is 1620 cm?
A)1.62 ohm
B)162 ohm
C)0.81 ohm
D)405 ohm

Accepted Answer

The resistance R is given by R = k * (l/d^2). First, find k using the initial conditions: 0.4 = k * (200/1^2), so k = 0.002. Then, use this k to find the new resistance: R = 0.002 * (1620/2^2) = 0.81 ohm.

Question 6

Find an equation of variation in which the following are true.
-$y$ varies jointly as $x$ and $z$, and $y = 175$ when $x = 5$ and $z = 7$
A) $y = \frac { 175 } { x Z }$
B) $y = 5 x z$
C) $y = 35 x$
D) $y = 7 x z$

Accepted Answer

The equation for joint variation is $y = kxz$. Given $y = 175$, $x = 5$, and $z = 7$, we can solve for $k$: $175 = k \cdot 5 \cdot 7$, which simplifies to $k = 5$. Thus, the equation is $y = 5xz$.

Question 7

Solve.
-The weight of a body above the surface of the earth varies inversely as the square of its distance from the center of the earth. What is the effect on the weight when the distance is multiplied by 2?
A)The weight is multiplied by 2.
B)The weight is divided by 4.
C)The weight is divided by 2.
D)The weight is multiplied by 4.

Accepted Answer

The weight varies inversely as the square of the distance, so if the distance is multiplied by 2, the weight is divided by $2^2 = 4$.

Question 8

Answer the question.
-Explain in your own words how to multiply rational expressions. (How would you explain this to a student in this class who was absent from class?).

Accepted Answer

The answer of Answer the question.
-Explain in your own words...

Question 9

Find an equation of variation in which the following are true.
-$y$ varies jointly as $x$ and $w$ and inversely as $z$, and $y = \frac { 9 } { 2 }$ when $x = 2 , w = 6$, and $z = 24$.
A) $y = \frac { 6 z } { x w }$
B) $y = 4 x w z$
C) $y = \frac { 6 x w } { z ^ { 2 } }$
D) $y = \frac { 9 x w } { z }$

Accepted Answer

The correct equation of variation is $y = \frac { 9 x w } { z }$ because it represents $y$ varying jointly as $x$ and $w$ (indicated by $xw$ in the numerator) and inversely as $z$ (indicated by $z$ in the denominator). The constants and variables are correctly aligned to satisfy the given conditions.

Question 10

Solve.
-The gravitational attraction A between two masses varies inversely as the square of the distance between them. The force of attraction is 9 lb when the masses are 2 ft apart, what is the attraction when the masses are 6 ft
Apart?
A)1 lb
B)2 lb
C)4 lb
D)3 lb

Accepted Answer

The gravitational attraction between two masses is given by the formula $A = \frac{k}{d^2}$, where $A$ is the attraction, $k$ is a constant, and $d$ is the distance between the masses. Given that the force of attraction is 9 lb when the distance is 2 ft, we can find $k$ by rearranging the formula to $k = A \cdot d^2 = 9 \cdot 2^2 = 36$. When the distance is 6 ft, the attraction is $A = \frac{36}{6^2} = \frac{36}{36} = 1$ lb.

Question 11

Find an equation of variation in which the following are true.
-$y$ varies jointly as $x$ and $p$ and inversely as the square of $s$, and $y = \frac { 9 } { 2 }$ when $x = 1 , p = 9$, and $s = 4$.
A) $y = \frac { 12x p ^ { 2 } } { s }$
B) $y = \frac { 32 x ^ { 2 } p } { s ^ { 2 } }$
C) $y = \frac { 8 x p } { s ^ { 2 } }$
D) $\mathrm { y } = 9 \mathrm { xps } ^ { 2 }$

Accepted Answer

The equation of variation where $y$ varies jointly as $x$ and $p$ and inversely as the square of $s$ is $y = k\frac{xp}{s^2}$, where $k$ is the constant of variation. Substituting the given values $y = \frac{9}{2}$, $x = 1$, $p = 9$, and $s = 4$ into the equation, we can solve for $k$ and find that $k = 8$, making the correct equation $y = \frac{8xp}{s^2}$.

Question 12

Find an equation of variation in which the following are true.
-$y$ varies directly as $x$ and inversely as $z$, and $y = 18$ when $x = 2$ and $z = 9$.
A) $y = \frac { 81 x } { z }$
B) $y = \frac { 85 x } { z }$
C) $y = 84 x z$
D) $y = \frac { 80 z } { x }$

Accepted Answer

The equation for $y$ varies directly as $x$ and inversely as $z$ is $y = k\frac{x}{z}$, where $k$ is the constant of variation. Substituting $x = 2$, $y = 18$, and $z = 9$ into the equation gives $18 = k\frac{2}{9}$. Solving for $k$ gives $k = 81$. Therefore, the equation is $y = \frac{81x}{z}$.

Question 13

Find an equation of variation in which the following are true.
-$y$ varies jointly as $x$ and $z$ and inversely as the product of $w$ and $p$, and $y = \frac { 45 } { 80 }$ when $x = 5 , z = 9 , w = 8$, and $\mathrm { p } = 5 .$
A) $y = \frac { 2 x z } { w p }$
B) $y = \frac { w p } { 2 x z }$
C) $y = \frac { 2 w p } { x z }$
D) $y = \frac { x z } { 2 w p }$

Accepted Answer

The answer of Find an equation of variation in which...

Question 14

Solve.
-Wind resistance or atmospheric drag tends to slow down moving objects. Atmospheric drag varies jointly as an object's surface area A and velocity v. If a car traveling at a speed of 40 mph with a surface area of $37 \mathrm { ft } ^ { 2 }$ experiences a drag of $236.8 \mathrm {~N}$ (Newtons), how fast must a car with $48 \mathrm { ft } ^ { 2 }$ of surface area travel in order to
A)66 mph
B)69 mph
C)71 mph
D)74 mph

Accepted Answer

The drag force $D$ is given by $D = kAv$, where $k$ is a constant, $A$ is the surface area, and $v$ is the velocity. Given $D_1 = 236.8$ N for $A_1 = 37 \mathrm{ft}^2$ and $v_1 = 40$ mph, we can find $k$ and then solve for $v_2$ given $A_2 = 48 \mathrm{ft}^2$. Since $D$ varies jointly with $A$ and $v$, we have $\frac{D_1}{A_1v_1} = \frac{D_2}{A_2v_2}$. Substituting the given values and solving for $v_2$ yields approximately 69 mph.

Question 15

Solve.
-The period of vibration P for a pendulum varies directly as the square root of the length L. If the period of vibration is 4 sec when the length is 64 inches, what is the period when L = 5.0625 inches?
A)4.25 sec
B)5 sec
C)1.125 sec
D)4.5 sec

Accepted Answer

The period of vibration $P$ is directly proportional to the square root of the length $L$, so $P = k\sqrt{L}$ where $k$ is a constant. Given $P = 4$ sec when $L = 64$ inches, we can find $k$ by substituting these values: $4 = k\sqrt{64}$, so $k = 4/8 = 0.5$. For $L = 5.0625$ inches, $P = 0.5\sqrt{5.0625} = 0.5 \times 2.25 = 1.125$ sec.

Question 16

Solve.
-The cost of stainless steel tubing varies jointly as the length and the diameter of the tubing. If a 5 foot length with diameter 2 inches costs $48.00 , how much will a 7 foot length with diameter 5 inches cost?
A)$168.00
B)$173.57
C)$173.30
D)$165.60

Accepted Answer

The cost varies jointly as the length and diameter, so C = k * L * D. For the first tubing, 48 = k * 5 * 2, giving k = 4.8. For the second tubing, C = 4.8 * 7 * 5 = 168.00.

Question 17

Solve.
-The force needed to keep a car from skidding on a curve varies jointly as the weight of the car and the square of the car's speed, and inversely as the radius of the curve. If a force of 3600 pounds is needed to keep an1800 pound car traveling at 20 mph from skidding on a curve of radius 600 feet, what force would be required To keep the same car from skidding on a curve of radius 550 feet at 60 mph? Round your answer to the nearest Pound of force?
A)35,213 pounds
B)35,345 pounds
C)35,377 pounds
D)35,915 pounds

Accepted Answer

The answer of Solve.
-The force needed to keep a car...

Question 18

Find an equation of variation in which the following are true.
-f varies jointly as $q ^ { 2 }$ and $h$, and $f = 96$ when $q = 4$ and $h = 3$.
A) $f = 2 q ^ { 2 } h$
B) $f = \frac { 1 } { 4608 } q ^ { 2 } h$
C) $f = 4608 q ^ { 2 } h$
D) $f = \frac { 1 } { 2 } q ^ { 2 } h$

Accepted Answer

The equation of variation is $f = kq^2h$. Substituting $f = 96$, $q = 4$, and $h = 3$ into the equation gives $96 = k(4)^2(3)$. Solving for $k$ gives $k = 2$, so the equation is $f = 2q^2h$.

Question 19

Solve.
-The volume of wood in a tree varies jointly as the height of the tree and the square of the distance around the tree trunk. If the volume of wood is 15.84 cubic feet when the height is 22 feet and the distance around the trunk
Is 3 feet, what is the volume of wood obtained from a tree that is 36 feet tall having a measurement of 4 feet
Around the trunk?
A)38.08 cubic feet
B)46.08 cubic feet
C)55.08 cubic feet
D)50.08 cubic feet

Accepted Answer

The volume of wood in a tree $V$ varies jointly as the height $h$ of the tree and the square of the distance $d$ around the tree trunk, which can be represented by the equation $V = k \cdot h \cdot d^2$, where $k$ is the constant of variation. Given $V = 15.84$ cubic feet when $h = 22$ feet and $d = 3$ feet, we can find $k$ by substituting these values into the equation: $15.84 = k \cdot 22 \cdot 3^2$. Solving for $k$ gives $k = 15.84 / (22 \cdot 9) = 0.08$.For a tree that is 36 feet tall with a distance of 4 feet around the trunk, we substitute $h = 36$, $d = 4$, and $k = 0.08$ into the equation to find the new volume $V$: $V = 0.08 \cdot 36 \cdot 4^2 = 0.08 \cdot 36 \cdot 16 = 46.08$ cubic feet.

Question 20

Solve.
-The intensity I of light varies inversely as the square of the distance D from the source. If the intensity of illumination on a screen 5 ft from a light is 4 foot candles, find the intensity on a screen 20 ft from the light.
Express the answer as a fraction. 
A) 4 foot-candles
B) $\frac { 1 } { 5 }$ foot-candle
C) $\frac { 1 } { 4 }$ foot-candle
D) 2 foot-candles

Accepted Answer

The intensity of light varies inversely as the square of the distance, so $I = \frac{k}{D^2}$, where $k$ is a constant. Given $I = 4$ foot-candles at $D = 5$ ft, we find $k = 4 \times 5^2 = 100$. At $D = 20$ ft, $I = \frac{100}{20^2} = \frac{100}{400} = \frac{1}{4}$ foot-candle.

Quiz 6: Rational Expressions and Equations

Answer the question. -For $x \neq 5$ , the rational expression $\frac { 8 ( x - 5 ) } { x - 5 }$ is equal to 8 . Can the same be said for $\frac { 8 x - 5 } { x - 5 }$ ? Explain why or why not.

Answer the question. -Explain in your own words how to divide rational expressions. (How would you explain this to a student in this class who was absent from class?).

Solve. -At a fixed temperature, the resistance R of a wire varies directly as the length l and inversely as the square of its diameter d. If the resistance is 0.4 ohm when the diameter is 1 mm and the length is 200 cm, what is the Resistance when the diameter is 2 mm and the length is 1620 cm?

Find an equation of variation in which the following are true. - $y$ varies jointly as $x$ and $z$ , and $y = 175$ when $x = 5$ and $z = 7$

Solve. -The weight of a body above the surface of the earth varies inversely as the square of its distance from the center of the earth. What is the effect on the weight when the distance is multiplied by 2?

Answer the question. -Explain in your own words how to multiply rational expressions. (How would you explain this to a student in this class who was absent from class?).

Find an equation of variation in which the following are true. - $y$ varies jointly as $x$ and $w$ and inversely as $z$ , and $y = \frac { 9 } { 2 }$ when $x = 2 , w = 6$ , and $z = 24$ .

Solve. -The gravitational attraction A between two masses varies inversely as the square of the distance between them. The force of attraction is 9 lb when the masses are 2 ft apart, what is the attraction when the masses are 6 ft Apart?

Find an equation of variation in which the following are true. - $y$ varies jointly as $x$ and $p$ and inversely as the square of $s$ , and $y = \frac { 9 } { 2 }$ when $x = 1 , p = 9$ , and $s = 4$ .

Find an equation of variation in which the following are true. - $y$ varies directly as $x$ and inversely as $z$ , and $y = 18$ when $x = 2$ and $z = 9$ .

Find an equation of variation in which the following are true. - $y$ varies jointly as $x$ and $z$ and inversely as the product of $w$ and $p$ , and $y = \frac { 45 } { 80 }$ when $x = 5 , z = 9 , w = 8$ , and $\mathrm { p } = 5 .$

Solve. -The period of vibration P for a pendulum varies directly as the square root of the length L. If the period of vibration is 4 sec when the length is 64 inches, what is the period when L = 5.0625 inches?

Solve. -The cost of stainless steel tubing varies jointly as the length and the diameter of the tubing. If a 5 foot length with diameter 2 inches costs $48.00 , how much will a 7 foot length with diameter 5 inches cost?

Find an equation of variation in which the following are true. -f varies jointly as $q ^ { 2 }$ and $h$ , and $f = 96$ when $q = 4$ and $h = 3$ .

Solve. -The intensity I of light varies inversely as the square of the distance D from the source. If the intensity of illumination on a screen 5 ft from a light is 4 foot candles, find the intensity on a screen 20 ft from the light. Express the answer as a fraction.

Quiz 6: Rational Expressions and Equations

Answer the question. -For x≠5x \neq 5x=5 , the rational expression 8(x−5)x−5\frac { 8 ( x - 5 ) } { x - 5 }x−58(x−5)​ is equal to 8 . Can the same be said for 8x−5x−5\frac { 8 x - 5 } { x - 5 }x−58x−5​ ? Explain why or why not.