Question 1

The force required to compress a spring varies directly as the change in the length of the spring. If a force of 18 pounds is required to compress a certain spring 4 inches, how much force is required to
Compress the spring 7 inches?

Accepted Answer

The force required to compress a spring varies directly with the change in length, meaning F = kx, where F is the force, x is the change in length, and k is a constant. Given F = 18 pounds for x = 4 inches, we find k = 18/4 = 4.5. For x = 7 inches, F = 4.5 * 7 = 31.5 pounds.

Question 2

Body mass index, or BMI, is used by physicians to assess a person's level of fatness. BMI varies directly as an individual's weight in pounds and inversely as the square of their height in inches. A person who weighs $179 \mathrm { lb }$ and is $72 \mathrm { in }$. tall has a BMI of 24 (rounded to the nearest whole number). Find the BMI of a person who weighs $155 \mathrm { lb }$ and who is 63 in tall. Round your answer to the nearest whole number.

Accepted Answer

The BMI formula is $\text{BMI} = \frac{\text{weight in pounds}}{(\text{height in inches})^2} \times 703$. Using the given information, we can calculate the BMI for the second person by adjusting for their weight and height. The calculation yields a BMI of approximately 27 when rounded to the nearest whole number.

Question 3

Match the inequality with its correct representation.
-$\{ \mathrm { x } \mid - 4 < \mathrm { x } < 8 \}$

Accepted Answer

The notation $\{ \mathrm { x } \mid - 4 < \mathrm { x } < 8 \}$ indicates all values of $x$ that are greater than -4 and less than 8, without including the endpoints -4 and 8. This is correctly represented by the interval notation $( - 4,8 )$, which also excludes -4 and 8.

Question 4

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 2 foot-candles, find the intensity on a screen 15 ft from the light.

Accepted Answer

The intensity of light varies inversely as the square of the distance, so if the distance triples (from 5 ft to 15 ft), the intensity decreases by a factor of $3^2 = 9$. Therefore, the new intensity is $\frac{2}{9}$ foot-candle.

Question 5

If $\mathrm { f }$ varies jointly as $\mathrm { q } ^ { 2 }$ and $h$, and $\mathrm { f } = 24$ when $\mathrm { q } = 2$ and $\mathrm { h } = 3$, find $\mathrm { h }$ when $\mathrm { f } = 160$ and $\mathrm { q } = 4$.

Accepted Answer

The formula for joint variation is $f = kq^2h$, where $k$ is the constant of variation. Given $f = 24$, $q = 2$, and $h = 3$, we can find $k$ by substituting these values into the formula: $24 = k(2)^2(3)$, which simplifies to $24 = 4k(3)$, and further to $k = 2$. To find $h$ when $f = 160$ and $q = 4$, substitute these values and the found $k$ into the formula: $160 = 2(4)^2h$, which simplifies to $160 = 32h$, and further to $h = 5$.

Question 6

The volume of wood, V, obtained from a tree varies jointly as the height, $h$, and the square of the girth (the distance around the tree trunk), g. If the volume is $38.4 \mathrm { ft } ^ { 3 }$ when the height is $30 \mathrm { ft }$ and the girth is $4 \mathrm { ft }$, what is the volume when the height is $50 \mathrm { ft }$ and the girth is $6 \mathrm { ft }$ ?

Accepted Answer

The answer of The volume of wood, V, obtained from...

Question 7

If $\mathrm { f }$ varies jointly as $\mathrm { q } ^ { 2 }$ and $h$, and $\mathrm { f } = - 81$ when $\mathrm { q } = 3$ and $\mathrm { h } = 3$, find $\mathrm { f }$ when $\mathrm { q } = 2$ and $h = 6$.

Accepted Answer

The answer of If $\mathrm { f }$ varies jointly...

Question 8

If $f$ varies jointly as $q ^ { 2 }$ and $h$, and $f = 36$ when $q = 2$ and $h = 3$, find $f$ when $q = 4$ and $h = 6$.

Accepted Answer

The answer of If $f$ varies jointly as \(q ^...

Question 9

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 $\mathrm { V } = 121.0$ in. ${ } ^ { 3 }$ when $\mathrm { T } = 110 ^ { \circ }$ and $\mathrm { P } = 10 \mathrm { lb } /$ in. 2 , what is the volume when $\mathrm { T } = 380 ^ { \circ }$ and $P = 15 \mathrm { lb } /$ in. $^ { 2 }$ ? (Round to the nearest tenth.)

Accepted Answer

The volume of a gas is given by $V = k \frac{T}{P}$, where $k$ is a constant. Using the initial conditions, we can find $k$ and then calculate the new volume with the given $T$ and $P$. The correct calculation leads to a volume of approximately $278.7 \, \text{in}^3$.

Question 10

The gravitational attraction A between two masses varies inversely as the square of the distance between them. If 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?

Accepted Answer

The gravitational attraction between two masses is given by the formula $A = \frac{k}{r^2}$, where $A$ is the attraction force, $k$ is a constant, and $r$ 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 $k = A \cdot r^2 = 9 \cdot 2^2 = 36$. When the distance is 6 ft, the attraction force is $A = \frac{36}{6^2} = \frac{36}{36} = 1$ lb.

Question 11

Match the inequality with its correct representation.
-$\{ x \mid - 6 \leq x < 6 \}$

Accepted Answer

The correct representation is B) $[ - 6,6 )$ because it includes $-6$ (as indicated by the square bracket) and goes up to, but does not include, $6$ (as indicated by the parenthesis). This matches the given inequality $- 6 \leq x < 6$.

Question 12

Match the inequality with its correct representation.
-$\{ x \mid - 8 < x \leq 7 \}$

Accepted Answer

The correct representation of the given inequality $\{ x \mid - 8 < x \leq 7 \}$ is $( - 8,7 ]$, indicating that $x$ is greater than -8 but less than or equal to 7. This is denoted by a parenthesis on the -8 (indicating that -8 is not included) and a bracket on the 7 (indicating that 7 is included).

Question 13

Match the inequality with its correct representation.
-$\{ x \mid x \leq - 1 \}$

Accepted Answer

The correct representation of the set $\{ x \mid x \leq - 1 \}$ is $( - \infty , - 1 ]$, indicating all numbers less than or equal to -1.

Question 14

Match the inequality with its correct representation.
-$\{ x \mid x > 4 \}$

Accepted Answer

The notation $\{ x \mid x > 4 \}$ describes all real numbers greater than 4, which is correctly represented by the interval notation $(4, \infty)$, indicating all numbers greater than 4 up to infinity.

Question 15

Match the inequality with its correct representation.
-$\{ x \mid - 5 \leq x \leq 9 \}$

Accepted Answer

The notation $\{ x \mid - 5 \leq x \leq 9 \}$ includes the endpoints -5 and 9, which is correctly represented by the closed interval $[ - 5,9 ]$, indicating that -5 and 9 are part of the set.

Question 16

The weight of a liquid varies directly as its volume V. If the weight of the liquid in a cubical container 4 cm on a side is 128 g, find the weight of the liquid in a cubical container 5 cm on a side.

Accepted Answer

The weight of the liquid varies directly with its volume. The volume of a cube is given by side^3. The volume of the first cube is 4^3 = 64 cm^3, and the weight is 128 g. The volume of the second cube is 5^3 = 125 cm^3. Using the direct variation formula (weight1/volume1 = weight2/volume2), we have 128/64 = weight2/125. Solving for weight2 gives 250 g.

Question 17

If $f$ varies jointly as $q ^ { 2 }$ and $h$, and $f = 24$ when $q = 2$ and $h = 3$, find $q$ when $f = 160$ and $h = 5$.

Accepted Answer

The joint variation equation is $f = kq^2h$. Given $f = 24$, $q = 2$, and $h = 3$, we find $k = 2$. Using $f = 160$ and $h = 5$, solve for $q$ to get $q = 4$.

Question 18

Over a specified distance, speed varies inversely with time. If Joe can drive from his house to his office in 40 minutes at an average speed of 40 miles per hour, what average speed is needed to
Drive the same distance in half an hour? If necessary, round your answer to the nearest tenth.

Accepted Answer

Speed varies inversely with time, so speed = k/time. If Joe can drive from his house to his office in 40 minutes at an average speed of 40 miles per hour, then 40 = k/40, so k = 1600. To drive the same distance in half an hour, the average speed needed is 1600/30 = 53.3 miles per hour.

Question 19

The time T necessary to make an enlargement of a photo negative varies directly as the area A of the enlargement. If 90 seconds are required to make a 3-by-5 enlargement, find the time required
For a 5-by-6 enlargement.

Accepted Answer

The area of the 3-by-5 enlargement is 15 square units, and it takes 90 seconds. The area of the 5-by-6 enlargement is 30 square units. Since time varies directly with area, the time required is (90/15) * 30 = 180 seconds.

Question 20

The period of vibration P for a pendulum varies directly as the square root of the length L. If the period of vibration is 2 sec when the length is 16 inches, what is the period when L = 6.25 inches?
(Round to the nearest hundredth.)

Accepted Answer

The relationship between the period of vibration (P) and the length (L) of a pendulum is given by $P = k\sqrt{L}$, where $k$ is a constant. Given $P = 2$ sec when $L = 16$ inches, we can find $k$ by substituting these values: $2 = k\sqrt{16}$ or $2 = 4k$, so $k = 0.5$. For $L = 6.25$ inches, $P = 0.5\sqrt{6.25} = 0.5 \times 2.5 = 1.25$ sec.

Quiz 7: The Basic Concepts of Algebra

The force required to compress a spring varies directly as the change in the length of the spring. If a force of 18 pounds is required to compress a certain spring 4 inches, how much force is required to Compress the spring 7 inches?

Match the inequality with its correct representation. - $\{ \mathrm { x } \mid - 4 < \mathrm { x } < 8 \}$

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 2 foot-candles, find the intensity on a screen 15 ft from the light.

If $\mathrm { f }$ varies jointly as $\mathrm { q } ^ { 2 }$ and $h$ , and $\mathrm { f } = 24$ when $\mathrm { q } = 2$ and $\mathrm { h } = 3$ , find $\mathrm { h }$ when $\mathrm { f } = 160$ and $\mathrm { q } = 4$ .

If $\mathrm { f }$ varies jointly as $\mathrm { q } ^ { 2 }$ and $h$ , and $\mathrm { f } = - 81$ when $\mathrm { q } = 3$ and $\mathrm { h } = 3$ , find $\mathrm { f }$ when $\mathrm { q } = 2$ and $h = 6$ .

If $f$ varies jointly as $q ^ { 2 }$ and $h$ , and $f = 36$ when $q = 2$ and $h = 3$ , find $f$ when $q = 4$ and $h = 6$ .

The gravitational attraction A between two masses varies inversely as the square of the distance between them. If 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?

Match the inequality with its correct representation. - $\{ x \mid - 6 \leq x < 6 \}$

Match the inequality with its correct representation. - $\{ x \mid - 8 < x \leq 7 \}$

Match the inequality with its correct representation. - $\{ x \mid x \leq - 1 \}$

Match the inequality with its correct representation. - $\{ x \mid x > 4 \}$

Match the inequality with its correct representation. - $\{ x \mid - 5 \leq x \leq 9 \}$

The weight of a liquid varies directly as its volume V. If the weight of the liquid in a cubical container 4 cm on a side is 128 g, find the weight of the liquid in a cubical container 5 cm on a side.

If $f$ varies jointly as $q ^ { 2 }$ and $h$ , and $f = 24$ when $q = 2$ and $h = 3$ , find $q$ when $f = 160$ and $h = 5$ .

Over a specified distance, speed varies inversely with time. If Joe can drive from his house to his office in 40 minutes at an average speed of 40 miles per hour, what average speed is needed to Drive the same distance in half an hour? If necessary, round your answer to the nearest tenth.

The time T necessary to make an enlargement of a photo negative varies directly as the area A of the enlargement. If 90 seconds are required to make a 3-by-5 enlargement, find the time required For a 5-by-6 enlargement.

The period of vibration P for a pendulum varies directly as the square root of the length L. If the period of vibration is 2 sec when the length is 16 inches, what is the period when L = 6.25 inches? (Round to the nearest hundredth.)

Quiz 7: The Basic Concepts of Algebra

The force required to compress a spring varies directly as the change in the length of the spring. If a force of 18 pounds is required to compress a certain spring 4 inches, how much force is required to Compress the spring 7 inches?

Match the inequality with its correct representation. - {x∣−4<x<8}\{ \mathrm { x } \mid - 4 < \mathrm { x } < 8 \}{x∣−4<x<8}

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 2 foot-candles, find the intensity on a screen 15 ft from the light.

If f\mathrm { f }f varies jointly as q2\mathrm { q } ^ { 2 }q2 and hhh , and f=24\mathrm { f } = 24f=24 when q=2\mathrm { q } = 2q=2 and h=3\mathrm { h } = 3h=3 , find h\mathrm { h }h when f=160\mathrm { f } = 160f=160 and q=4\mathrm { q } = 4q=4 .

If f\mathrm { f }f varies jointly as q2\mathrm { q } ^ { 2 }q2 and hhh , and f=−81\mathrm { f } = - 81f=−81 when q=3\mathrm { q } = 3q=3 and h=3\mathrm { h } = 3h=3 , find f\mathrm { f }f when q=2\mathrm { q } = 2q=2 and h=6h = 6h=6 .

If fff varies jointly as q2q ^ { 2 }q2 and hhh , and f=36f = 36f=36 when q=2q = 2q=2 and h=3h = 3h=3 , find fff when q=4q = 4q=4 and h=6h = 6h=6 .

The gravitational attraction A between two masses varies inversely as the square of the distance between them. If 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?

Match the inequality with its correct representation. - {x∣−6≤x<6}\{ x \mid - 6 \leq x < 6 \}{x∣−6≤x<6}

Match the inequality with its correct representation. - {x∣−8<x≤7}\{ x \mid - 8 < x \leq 7 \}{x∣−8<x≤7}

Match the inequality with its correct representation. - {x∣x≤−1}\{ x \mid x \leq - 1 \}{x∣x≤−1}

Match the inequality with its correct representation. - {x∣x>4}\{ x \mid x > 4 \}{x∣x>4}

Match the inequality with its correct representation. - {x∣−5≤x≤9}\{ x \mid - 5 \leq x \leq 9 \}{x∣−5≤x≤9}

The weight of a liquid varies directly as its volume V. If the weight of the liquid in a cubical container 4 cm on a side is 128 g, find the weight of the liquid in a cubical container 5 cm on a side.

If fff varies jointly as q2q ^ { 2 }q2 and hhh , and f=24f = 24f=24 when q=2q = 2q=2 and h=3h = 3h=3 , find qqq when f=160f = 160f=160 and h=5h = 5h=5 .

Over a specified distance, speed varies inversely with time. If Joe can drive from his house to his office in 40 minutes at an average speed of 40 miles per hour, what average speed is needed to Drive the same distance in half an hour? If necessary, round your answer to the nearest tenth.

The time T necessary to make an enlargement of a photo negative varies directly as the area A of the enlargement. If 90 seconds are required to make a 3-by-5 enlargement, find the time required For a 5-by-6 enlargement.

The period of vibration P for a pendulum varies directly as the square root of the length L. If the period of vibration is 2 sec when the length is 16 inches, what is the period when L = 6.25 inches? (Round to the nearest hundredth.)

Match the inequality with its correct representation. - $\{ \mathrm { x } \mid - 4 < \mathrm { x } < 8 \}$

If $\mathrm { f }$ varies jointly as $\mathrm { q } ^ { 2 }$ and $h$ , and $\mathrm { f } = 24$ when $\mathrm { q } = 2$ and $\mathrm { h } = 3$ , find $\mathrm { h }$ when $\mathrm { f } = 160$ and $\mathrm { q } = 4$ .

If $\mathrm { f }$ varies jointly as $\mathrm { q } ^ { 2 }$ and $h$ , and $\mathrm { f } = - 81$ when $\mathrm { q } = 3$ and $\mathrm { h } = 3$ , find $\mathrm { f }$ when $\mathrm { q } = 2$ and $h = 6$ .

If $f$ varies jointly as $q ^ { 2 }$ and $h$ , and $f = 36$ when $q = 2$ and $h = 3$ , find $f$ when $q = 4$ and $h = 6$ .

Match the inequality with its correct representation. - $\{ x \mid - 6 \leq x < 6 \}$

Match the inequality with its correct representation. - $\{ x \mid - 8 < x \leq 7 \}$

Match the inequality with its correct representation. - $\{ x \mid x \leq - 1 \}$

Match the inequality with its correct representation. - $\{ x \mid x > 4 \}$

Match the inequality with its correct representation. - $\{ x \mid - 5 \leq x \leq 9 \}$

If $f$ varies jointly as $q ^ { 2 }$ and $h$ , and $f = 24$ when $q = 2$ and $h = 3$ , find $q$ when $f = 160$ and $h = 5$ .