The force F (in pounds) needed on a wrench handle to loosen a certain bolt varies inversely with the length L (in inches) of the handle. A force of 45 pounds is needed when the handle is 5 inches long. If a person needs 25 pounds of force to loosen the bolt, estimate the length of the wrench handle. Round answer to two decimal places if necessary.
45 * 5 = 225
25 * L = 225
To solve this problem, we can use the inverse variation formula:
F = k/L
where F is the force (in pounds), L is the length of the wrench handle (in inches), and k is the constant of variation.
From the problem, we know that when F = 45 pounds, L = 5 inches. We can use this information to find k:
45 = k/5
To solve for k, we can multiply both sides of the equation by 5:
k = 45 * 5
k = 225
Now that we have the constant of variation, k, we can use it to find the length, L, when F = 25 pounds:
25 = 225 / L
To solve for L, we can multiply both sides of the equation by L:
25L = 225
Next, divide both sides of the equation by 25:
L = 225 / 25
L = 9
Therefore, the estimated length of the wrench handle needed to apply a force of 25 pounds is 9 inches.
To solve this problem, we can use the concept of inverse variation. Inverse variation refers to a mathematical relationship where one variable is inversely proportional to another variable. In this case, the force needed on a wrench handle and the length of the handle have an inverse variation relationship.
We can start by writing an equation to represent this relationship. Let F be the force needed on the wrench handle, and let L be the length of the handle. According to the problem, the force F varies inversely with the length L, so we can express this as:
F ∝ 1/L
Using the proportional symbol (∝) means that F is directly proportional to the reciprocal of L.
We can introduce a constant of variation k into the equation:
F = k/L
To find the value of k, we can use the given information. The problem states that a force of 45 pounds is needed when the handle is 5 inches long. Therefore, we can substitute these values into the equation:
45 = k/5
To solve for k, we can multiply both sides of the equation by 5:
225 = k
Now that we have the value of k, we can write the equation in terms of F and L:
F = 225/L
Next, we are asked to estimate the length of the wrench handle if a person needs 25 pounds of force to loosen the bolt. We can use the equation and substitute 25 for F:
25 = 225/L
Solving for L, we can multiply both sides of the equation by L:
25L = 225
Dividing both sides of the equation by 25, we get:
L = 9
Therefore, the estimated length of the wrench handle is 9 inches.
In summary, to estimate the length of the wrench handle, we used the concept of inverse variation and the given information to form an equation. Then we solved for the length L by substituting the given force value of 25 pounds into the equation. The estimated length of the wrench handle is 9 inches.