Question 1

True or false? The indicated graph does not represent a function of x .

Accepted Answer

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Question 2

Express the sentence as a formula. B varies directly with m and inversely with n .

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Question 3

The pressure of a certain amount of gas is directly proportional to the temperature (measured in degrees Kelvin) and inversely proportional to the volume. A sample of gas at a pressure of 1 atmosphere occupies a volume of 1 cubic meter at a temperature of 277 Kelvin. When heated, the gas expands to twice its volume, but the pressure remains constant. To what temperature is it heated?

Accepted Answer

The relationship between pressure, volume, and temperature for a given amount of gas is described by the ideal gas law, which can be expressed as $P \propto \frac{T}{V}$, where $P$ is the pressure, $T$ is the temperature in Kelvin, and $V$ is the volume. Given that the volume doubles and the pressure remains constant, the temperature must also double to maintain the relationship. Therefore, the new temperature is $277 \times 2 = 554$ Kelvin.

Question 4

Express the formula in words. k is the constant of variation.

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Question 5

Graph the function and find its domain and range. f ( x ) = | x | - 2

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Question 6

Graph the function and find its domain and range. f ( x ) = -2| x | + 5

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Question 7

Assume that all variables represent positive numbers. Assume that  varies jointly with  and  . If  when  and  , find  when  and  .

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Question 8

Determine whether the indicated graph represents a function of x . Please enter yes or no .

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Question 9

The cost of drilling a water well is jointly proportional to the length and diameter of the steel casing. If a 29-foot well using 3-inch casing cost $957, find the cost of a 35-foot well using 6-inch casing.

Accepted Answer

The cost is jointly proportional to the length and diameter, so we use the formula: Cost = k * length * diameter. First, find k using the given data: 957 = k * 29 * 3. Solving for k gives k = 11. Then, use k to find the cost of the new well: Cost = 11 * 35 * 6 = $2,310.

Question 10

The deflection of a beam is inversely proportional to its width and the cube of its depth. If the deflection is 3 inches when the width is 4 inches and the depth is 3 inches, find the deflection when the width is 3 inches and the depth is 4 inches.

Accepted Answer

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Question 11

True or false? The indicated graph represents a function of x .

Accepted Answer

The graph passes the vertical line test, meaning that for any vertical line drawn on the graph, it intersects the graph at most once, which indicates that the graph represents a function of x.

Question 12

The time it takes a truck to go a certain distance varies inversely with its rate of speed. If a certain trip takes 3 hours when the driver goes 30 mph, how long will the trip take when the driver goes 60 mph?

Accepted Answer

The relationship between time and speed in this context is inversely proportional, meaning as one increases, the other decreases. The formula for this relationship is $T_1 \times S_1 = T_2 \times S_2$, where $T_1$ and $S_1$ are the original time and speed, and $T_2$ and $S_2$ are the new time and speed. Plugging in the given values: $3 \times 30 = T_2 \times 60$. Solving for $T_2$ gives $T_2 = 1.5$ hours.

Question 13

Graph the function and find its domain and range.

Accepted Answer

Without information about the actual function, we cannot determine the domain and range for sure. However, based solely on the fact that option C has a non-vertical asymptote at x=0 (indicating that the function approaches both positive and negative infinity), and the other options do not have this characteristic, we can identify C as the best choice. The domain of option C is all real numbers, since there are no restrictions given, and the range is also all real numbers, since the function approaches both positive and negative infinity.

Question 14

The distance that a bus can go without refueling varies directly with the number of gallons of gasoline in the tank. If the bus can go 247 miles on a full tank of gas (13 gallons), how far can it go on 8 gallons of gas?

Accepted Answer

The distance varies directly with the gallons of gasoline, so we use the formula: distance = (miles per gallon) * gallons. Given 247 miles for 13 gallons, the bus travels 247/13 = 19 miles per gallon. For 8 gallons, it can go 19 * 8 = 152 miles.

Question 15

Express the sentence as a formula. Assume that k is a constant of variation. A farmer's harvest h varies directly with  , the number of acres he plants.

Accepted Answer

The formula for direct variation is y=kx, where y is the dependent variable (harvest in this case), x is the independent variable (number of acres planted), and k is the constant of variation. Using the given information, we can write the formula as h=k(11ec187c_74ef_da5b_b8be_011b250b2a3f_TBX8672_11). Therefore, choice B is the correct answer.

Question 16

The volume of a rectangular solid varies jointly with its length, width, and height. If the length is tripled, the width is tripled, and the height is quadrupled, by what factor is the volume multiplied?

Accepted Answer

Let V be the original volume, and let L, W, and H be the original length, width, and height, respectively. We know that V is jointly proportional to L, W, and H, so we can write:
V = kLWH
where k is some constant of variation.

If we triple the length, width, and height, we get a new volume, call it V', given by:
V' = k(3L)(3W)(4H) = 36kLWH

So the new volume is 36 times the original volume, which means the volume is multiplied by a factor of 36. Therefore, the answer is D, 36.

Question 17

For a fixed rate and principal, the interest earned in a bank account paying simple interest varies directly with the length of time the principal is left on deposit. If an investment of $7,000 earns $700 in 2 years, how much will it earn in 8 years?

Accepted Answer

The interest earned is directly proportional to the time. Since $700 is earned in 2 years, in 8 years (which is 4 times longer), the interest will be 4 times $700, which equals $2,800.

Question 18

Express the sentence as a formula. Assume that  is a constant of variation. For a fixed area, the length  of a rectangle varies inversely with its width  .

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Quiz 3: Graphs; Equations of Lines; Functions; Variation