A)how do I find an inverse variaton equation that model:
Length(in.) 4 6 8 9 10
Breaking weight(pennies)24 16 13 11 9
B)Explain how your equation shows that breaking weight decreases as length increases. Is this pattern reasonable for this situation? Explain.
To find an inverse variation equation that models the given data, follow these steps:
Step 1: Note down the given data points in the form of (x, y).
Length(in.) Breaking weight(pennies)
4 24
6 16
8 13
9 11
10 9
Step 2: Inverse variation equations can be represented as y = k/x, where k is a constant.
Step 3: Substitute the given data points into the equation to find the value of the constant k. You can choose any data point for substitution. Let's use the first point, (4, 24):
24 = k/4
Step 4: Solve the equation for k. Multiply both sides by 4:
k = 24 * 4
k = 96
Step 5: Write the inverse variation equation in the form y = k/x using the value of k:
Breaking weight = 96/Length
So, the inverse variation equation that models the given data is Breaking weight = 96/Length.
Now, to explain how the equation shows that breaking weight decreases as length increases:
In the inverse variation equation y = k/x, as the value of x (length) increases, the value of y (breaking weight) decreases. This means that as the length of the object increases, the breaking weight decreases, which is consistent with the observation in the given data.
In this situation, it is reasonable for breaking weight to decrease as length increases because longer objects tend to have a greater likelihood of bending or breaking due to their increased weight distribution. As the length of the object increases, it becomes more difficult to distribute the weight evenly, resulting in a decrease in breaking weight.
A) To find an inverse variation equation that models the relationship between length (in.) and breaking weight (pennies), you need to first understand the concept of inverse variation. In an inverse variation, as one variable increases, the other variable decreases, and vice versa. The general form of an inverse variation equation is y = k/x, where y is the dependent variable, x is the independent variable, and k is a constant.
To find the equation, we can use the data given in the problem. We have two columns: Length (in.) and Breaking weight (pennies).
Length (in.): 4 6 8 9 10
Breaking weight (pennies): 24 16 13 11 9
In order to have an inverse variation equation, we need to find a constant, k. We can do this by multiplying the length and breaking weight values:
4 * 24 = 96
6 * 16 = 96
8 * 13 = 104
9 * 11 = 99
10 * 9 = 90
As you can see, the product of length and breaking weight is approximately the same for each pair of values. Choosing any pair, let's say (4, 24), we can set up the equation y = k/x using the product:
24 = k/4
To solve for k, we can multiply both sides by 4:
4 * 24 = k
k = 96
Therefore, the inverse variation equation that models this relationship is:
Breaking weight (pennies) = 96/Length (in.)
B) The equation Breaking weight (pennies) = 96/Length (in.) shows that as the length increases, the breaking weight decreases. This is because inverse variation states that as one variable increases, the other variable decreases. In this case, as the length of the object increases, the breaking weight decreases according to the inverse variation equation.
This pattern of breaking weight decreasing as length increases is reasonable for this situation. In real-life scenarios, objects that are longer tend to be thinner, which means they have less material to support the weight. As a result, longer objects are generally weaker and more prone to breaking under the same weight compared to shorter objects with the same material composition. Thus, the inverse relationship between length and breaking weight aligns with our intuition and observations.
A neat math trick on inverse relationships is this trick:
IF A is inversley related to B, then
A= constant/B where constant is the constant of proportionality.
Then multiplying both sides by B, then
A*B= constant.
So check your data, if length*Weight stays constant, it is inversely related.
I will do a couple for you.
4*24= ...96
6*16= ...96
check the rest.