Math / Algebra

QuestionThe ends of a bar have to be machined down until the bar is 3 m long. After $5 \%$ of the length had been removed, it was found that $0.5 \%$ of the new length still had to come off. What length was removed at each machining and what was the original length of the bar?

Studdy Solution

STEP 1

1. The final length of the bar is 3 meters.
2. Initially, 5% of the original length was removed.
3. After the first removal, 0.5% of the new length still needed to be removed to reach the final length.
4. We need to find the length removed at each machining and the original length of the bar.

STEP 2

1. Define variables for the original length and the lengths removed.
2. Set up equations based on the problem description.
3. Solve the equations to find the original length and the lengths removed.

STEP 3

Define variables. Let $L$ be the original length of the bar. Let $R_1$ be the length removed in the first machining, and $R_2$ be the length removed in the second machining.

STEP 4

Set up equations based on the problem description.
The first machining removed 5% of the original length: $R_1 = 0.05L$
After the first machining, the length of the bar is $L - R_1$ .
The second machining removed 0.5% of the new length, and the final length is 3 meters: $R_2 = 0.005(L - R_1)$
The total length removed is the sum of $R_1$ and $R_2$ , and the final length is 3 meters: $L - R_1 - R_2 = 3$

STEP 5

Substitute $R_1$ and $R_2$ into the equation and solve for $L$ .
Substitute $R_1 = 0.05L$ into $R_2$ : $R_2 = 0.005(L - 0.05L)$ $R_2 = 0.005(0.95L)$ $R_2 = 0.00475L$
Substitute $R_1$ and $R_2$ into the total length equation: $L - 0.05L - 0.00475L = 3$ $L(1 - 0.05 - 0.00475) = 3$ $L(0.94525) = 3$
Solve for $L$ : $L = \frac{3}{0.94525}$ $L \approx 3.173 \, \text{meters}$

STEP 6

Calculate the lengths removed at each machining.
Calculate $R_1$ : $R_1 = 0.05 \times 3.173$ $R_1 \approx 0.15865 \, \text{meters}$
Calculate $R_2$ : $R_2 = 0.00475 \times 3.173$ $R_2 \approx 0.01507 \, \text{meters}$
The original length of the bar was approximately $3.173 \, \text{meters}$ . The length removed at the first machining was approximately $0.15865 \, \text{meters}$ . The length removed at the second machining was approximately $0.01507 \, \text{meters}$ .

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QuestionThe ends of a bar have to be machined down until the bar is 3 m long. After $5 \%$ of the length had been removed, it was found that $0.5 \%$ of the new length still had to come off. What length was removed at each machining and what was the original length of the bar?

Studdy Solution

STEP 1

1. The final length of the bar is 3 meters.
2. Initially, 5% of the original length was removed.
3. After the first removal, 0.5% of the new length still needed to be removed to reach the final length.
4. We need to find the length removed at each machining and the original length of the bar.

STEP 2

1. Define variables for the original length and the lengths removed.
2. Set up equations based on the problem description.
3. Solve the equations to find the original length and the lengths removed.

STEP 3

Define variables. Let $L$ be the original length of the bar. Let $R_1$ be the length removed in the first machining, and $R_2$ be the length removed in the second machining.

STEP 4

STEP 5

STEP 6

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QuestionThe ends of a bar have to be machined down until the bar is 3 m long. After 5%5 \%5% of the length had been removed, it was found that 0.5%0.5 \%0.5% of the new length still had to come off. What length was removed at each machining and what was the original length of the bar?

Studdy Solution

STEP 1

1. The final length of the bar is 3 meters.2. Initially, 5% of the original length was removed.3. After the first removal, 0.5% of the new length still needed to be removed to reach the final length.4. We need to find the length removed at each machining and the original length of the bar.

STEP 2

1. Define variables for the original length and the lengths removed.2. Set up equations based on the problem description.3. Solve the equations to find the original length and the lengths removed.

STEP 3

Define variables. Let L L L be the original length of the bar. Let R1 R_1 R1​ be the length removed in the first machining, and R2 R_2 R2​ be the length removed in the second machining.

STEP 4

STEP 5

STEP 6

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionThe ends of a bar have to be machined down until the bar is 3 m long. After $5 \%$ of the length had been removed, it was found that $0.5 \%$ of the new length still had to come off. What length was removed at each machining and what was the original length of the bar?

1. The final length of the bar is 3 meters.
2. Initially, 5% of the original length was removed.
3. After the first removal, 0.5% of the new length still needed to be removed to reach the final length.
4. We need to find the length removed at each machining and the original length of the bar.

1. Define variables for the original length and the lengths removed.
2. Set up equations based on the problem description.
3. Solve the equations to find the original length and the lengths removed.

Define variables. Let $L$ be the original length of the bar. Let $R_1$ be the length removed in the first machining, and $R_2$ be the length removed in the second machining.