Self-study (Variation) Home Learning

Explanation
The answer c,C shows that A is inversely proportional to B, where k is a constant. Inverse proportionality means that as one variable increases, the other variable decreases, and vice versa. The lowercase c represents a constant value for A, while the uppercase C represents a constant value for B. This indicates that as the constant value for A increases, the constant value for B decreases, and vice versa, confirming the inverse relationship between A and B.

Explanation
The answer d,D indicates that y varies directly as the square root of x. In mathematical terms, this means that as x increases, y also increases, and the relationship between y and x can be represented by the equation y = k√x, where k is a constant. The lowercase "d" represents the direct variation symbol, indicating the relationship between y and x, while the uppercase "D" represents the square root symbol, indicating that y varies directly with the square root of x.

Explanation
The answer "c,C" suggests that the variables y and x are inversely proportional to each other. This means that as the value of x increases, the value of y decreases, and vice versa. The lowercase "c" and uppercase "C" represent different values of x and y that follow this inverse relationship.

Explanation
The lowercase "b" and uppercase "B" in the given options indicate that y varies directly as the square of x. In mathematics, when two variables vary directly, it means that as one variable increases, the other variable also increases at a constant ratio. In this case, y varies directly as the square of x, which means that when x is squared, the resulting value is directly proportional to y. Therefore, both options "b" and "B" correctly represent the direct variation between y and the square of x.

Explanation
The correct answer is d, D. Inverse variation means that as one variable increases, the other variable decreases, and vice versa. In this case, y varies inversely as the square root of x, which means that as x increases, y decreases, and as x decreases, y increases. The answer d, D suggests that as x increases, y decreases, and as x decreases, y increases, which aligns with the concept of inverse variation.

Explanation
The correct answer is c,C. This means that the relationship between A, x, and y can be represented as A = k(x^3/y), where k is a constant. This indicates that as x increases, A will increase cubically, and as y increases, A will decrease inversely. The lowercase "c" represents the direct variation with x, and the uppercase "C" represents the inverse variation with y.

Explanation
The correct answer is b,B. This means that the statement "P varies inversely as the square of (x - 5) and directly as the cube root of y" is true. In other words, as (x - 5) increases, P decreases, and as y increases, P also increases. The constant k represents the proportionality constant that relates P to (x - 5) and y.

Explanation
The given problem states that the amount of light received, L, varies inversely as the square of the distance, d. This means that as the distance increases, the amount of light received decreases. The problem also provides a specific scenario where L = 8 when d = 3. Using this information, we can set up an equation: L = k/d^2, where k is a constant. Substituting the given values, we get 8 = k/3^2, which simplifies to k = 72. To find the value of d when L = 288, we can plug it into the equation: 288 = 72/d^2. Solving for d, we get d = 0.5 or 1/2.

Explanation
The given information states that y is directly proportional to the cube of (x - 2). This means that as (x - 2) increases, y will increase as well. By substituting the given values, we can find the constant of proportionality, which is 1 in this case. Therefore, we can set up the equation y = k(x - 2)^3, where k is the constant. Plugging in y = 72 and solving for x, we get (72) = 1(x - 2)^3. Simplifying the equation, we find that (x - 2)^3 = 72. Taking the cube root of both sides, we find that x - 2 = 4. Solving for x, we get x = 6.

Explanation
In this question, we are given that y varies jointly directly as the cube of x and inversely as the square root of (x - 2). This can be represented as y = k * (x^3) / sqrt(x - 2), where k is the constant of proportionality. To find the value of k, we can substitute the given values of y and x in the equation. When y = 12 and x = 3, we have 12 = k * (3^3) / sqrt(3 - 2). Simplifying this equation, we get k = 12 * sqrt(1) = 12. Now, we can substitute the value of k and x in the equation to find y when x = 6. y = 12 * (6^3) / sqrt(6 - 2) = 12 * 216 / sqrt(4) = 12 * 216 / 2 = 48. Therefore, y = 48 when x = 6.

Explanation
If twelve men can paint the wall in 14 days, it means that the rate at which they paint is 1/14th of the wall per day. To find out how long it takes twenty-one men to paint the same wall, we can use the formula: time = work / rate. Since the work is the same (painting the entire wall), and the rate is 1/14th of the wall per day, we can plug in the values to find the time. Therefore, it will take twenty-one men 8 days to paint the same wall.

Explanation
After working for 2 hours, the 15 men have completed 2/8 or 1/4 of the work. Therefore, 3/4 of the work remains to be completed. Since 18 men are now working, the work will be completed in 3/4 of the time it took for 15 men to complete the entire work. Since 15 men took 8 hours to complete the work, 18 men will take (3/4) * 8 = 6 hours to complete the remaining work.

Explanation
Since g is directly proportional to (h + 1)2, we can write the equation as g = k(h + 1)2, where k is the constant of proportionality. To find the value of k, we can substitute the given values of g and h into the equation. When g = 4 and h = 3, we have 4 = k(3 + 1)2. Simplifying this equation gives us 4 = k(4)2, which further simplifies to 4 = 16k. Solving for k, we find k = 4/16 = 1/4. Now, we can substitute the value of k into the equation and find the value of g when h = -2. g = (1/4)(-2 + 1)2 = (1/4)(-1)2 = 1/4. Therefore, the value of g when h = -2 is 1/4.

Explanation
The speed of a car is proportional to the square root of the length of the skid. In this case, the car skids 25 meters at a speed of 20 m/s. To find the speed of a car that skids 100 meters, we can set up a proportion. The square root of the length of the skid is directly proportional to the speed. Therefore, if the length of the skid is 4 times larger (100/25), the speed should also be 4 times larger. Since the initial speed is 20 m/s, the speed of a car that skids 100 meters would be 4 times larger, which is 80 m/s.

Explanation
If 15 men can paint a house in 8 days, it means that the total work required to paint the house can be completed by 15 men in 8 days. To find out how long it would take 4 men to paint the same house, we can use the concept of man-days. The total man-days required to paint the house is 15 men * 8 days = 120 man-days. If we divide this by the number of men (4), we get 120 man-days / 4 men = 30 days. Therefore, it would take 4 men 30 days to paint the same house.