1. If y varies directly as x, and y=30 when x=4, find the constant of proportionality.
2. If y varies inversely as x, and y=2 when x=10, find x when y=4.
3. If a varies jointly as b and c, and a=-10, when b=12 and c=2, find a when b=8 and c=1/2.
1. y=kx plug in your # and solve for k
2. y=k/x solve for k and then plug in your y=4 to find x with the k value you found.
3. I forgot how to do this one.. it's been years lol.
50
1. To find the constant of proportionality, we can use the formula for direct variation, which is y = kx. Here, y represents the dependent variable, x represents the independent variable, and k represents the constant of proportionality.
Given that y = 30 when x = 4, we can substitute these values into the equation to solve for k:
30 = k * 4
Dividing both sides of the equation by 4, we get:
k = 30 / 4
k = 7.5
Therefore, the constant of proportionality is 7.5.
2. To find x when y = 4 in an inverse variation problem, we can use the formula y = k/x. Here, y represents the dependent variable, x represents the independent variable, and k represents the constant of proportionality.
Given that y = 2 when x = 10, we can substitute these values into the equation to solve for k:
2 = k / 10
Multiplying both sides of the equation by 10, we get:
20 = k
Now, we can substitute the value of k into the inverse variation formula to find x when y = 4:
4 = 20 / x
Multiplying both sides of the equation by x, we get:
4x = 20
Dividing both sides of the equation by 4, we get:
x = 20 / 4
x = 5
Therefore, when y = 4, x = 5 in this inverse variation problem.
3. To find a when b = 8 and c = 1/2 in a joint variation problem, we can use the formula a = kbc. Here, a represents the dependent variable, b and c represent the independent variables, and k represents the constant of proportionality.
Given that a = -10 when b = 12 and c = 2, we can substitute these values into the equation to solve for k:
-10 = k * 12 * 2
Simplifying the equation, we get:
-10 = 24k
Dividing both sides of the equation by 24, we get:
k = -10 / 24
k = -5/12
Now, we can substitute the value of k, b = 8, and c = 1/2 into the joint variation formula to find a:
a = (-5/12) * 8 * (1/2)
Simplifying the equation, we get:
a = (-5/12) * 4
a = -20/12
a = -5/3
Therefore, when b = 8 and c = 1/2, a = -5/3 in this joint variation problem.
To solve these problems, we need to understand the concept of direct variation, inverse variation, and joint variation.
1. In direct variation, if y varies directly as x, it means that y and x are proportional to each other. Mathematically, it can be represented as y = kx, where k is the constant of proportionality.
To find the constant of proportionality, we can use the given information:
y = 30 when x = 4
Substituting these values into the equation, we get:
30 = k * 4
To isolate k, we divide both sides of the equation by 4:
k = 30/4
k = 7.5
So, the constant of proportionality is 7.5.
2. In inverse variation, if y varies inversely as x, it means that their product is a constant. Mathematically, it can be represented as y = k/x, where k is the constant of variation.
To find x when y = 4, we can use the given information:
y = 2 when x = 10
Substituting these values into the equation, we get:
2 = k/10
To isolate k, we multiply both sides of the equation by 10:
20 = k
Now that we have the constant of variation, we can find x when y = 4:
4 = 20/x
To isolate x, we divide both sides of the equation by 4:
x = 20/4
x = 5
So, when y = 4, x = 5.
3. In joint variation, if a varies jointly as b and c, it means that a is directly proportional to both b and c. Mathematically, it can be represented as a = k * b * c, where k is the constant of proportionality.
To find a when b = 8 and c = 1/2, we can use the given information:
a = -10 when b = 12 and c = 2
Substituting these values into the equation, we get:
-10 = k * 12 * 2
To isolate k, we divide both sides of the equation by 24:
k = -10/24
k = -5/12
Now that we have the constant of proportionality, we can find a when b = 8 and c = 1/2:
a = (-5/12) * 8 * (1/2)
a = -40/24
a = -5/3
So, when b = 8 and c = 1/2, a = -5/3.