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.
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.
45 * 5 = 225
25 * L = 225