1. The length of a rectangle with a constant area varies inversely as its width. The length of this rectangle is 8in when its width is 3in. Find the length when the width is 4in.
2. The number of people working on a certain job varies inversely as the number of days needed to finsih the job. If it takes 10 carpenters to finish a house in 8 days, in how many days will 4 carpenters finish it?
3. Work varies jointly as force and distance. How much work is done when a 30 N object is raised to a height of 15m?
What have you done so far?
1. To solve this problem, we need to apply the inverse variation formula. Inverse variation is a relationship where one variable increases while the other decreases, and their product remains constant.
Let's call the length of the rectangle "L" and the width "W." We know that L and W are inversely proportional, so we can express this with the equation L = k/W, where k is the constant of variation.
To find the value of k, we can use the given information. When the length is 8 inches and the width is 3 inches, we substitute these values into the equation: 8 = k/3.
To solve for k, we can cross-multiply: 8 * 3 = k, which simplifies to k = 24.
Now that we have the value of k, we can find the length when the width is 4 inches. We substitute W = 4 into the equation: L = 24/4.
Simplifying further, we get L = 6 inches. So, when the width is 4 inches, the length of the rectangle is 6 inches.
2. This problem also involves inverse variation. We are given that the number of people (P) working on a job varies inversely with the number of days (D) needed to finish the job.
Using the inverse variation formula, we can write the equation P = k/D, where k is the constant of variation.
Given that 10 carpenters can finish the house in 8 days, we can substitute P = 10 and D = 8 into the equation: 10 = k/8.
To find k, we can cross-multiply: 10 * 8 = k, which simplifies to k = 80.
Now we can use this value of k to determine how many days it will take 4 carpenters to finish the job. We substitute P = 4 into the equation: 4 = 80/D.
Solving for D, we have D = 80/4 = 20 days. Therefore, it will take 4 carpenters 20 days to finish the job.
3. Work varies jointly as force and distance. This means that the amount of work (W) is directly proportional to both the force (F) and the distance (D) covered.
We can write this relationship as an equation: W = k * F * D, where k is the constant of proportionality.
To find the value of k, we need to use the given information. When a 30 N object is raised to a height of 15 m, we substitute F = 30 and D = 15 into the equation: W = k * 30 * 15.
Since we don't have a specific value for W, we can consider it as an unknown constant in this case.
Now, we can solve for k by dividing both sides of the equation by (30 * 15): W / (30 * 15) = k.
Simplifying further, we get k = W / 450.
Therefore, the amount of work done depends on the value of W. Without specifying the value for W, we cannot determine the exact amount of work done in this situation.
I only did 1 and 3 but I don't know if its correct.
1. L = a/w ===== 8 = a/3
(3(8) = (a/3)(3)
24 = a
L = 24/4 = 6
3. W = FD ====== (60N)(15m) = 900 Nm