Question

When a part is made by pressing together powdered metal, no metal is wasted. In contrast, metal is typically scrapped after a metal part is cut from a solid metal plate. If a circular part with a diameter of \(10.0 \mathrm{~cm}\) is made from an original shape of a square with a side length of \(10.0 \mathrm{~cm},\) calculate the percentage of the metal that is thrown away as scrap. If the circular shape also has a circular hole with a diameter of \(6.0 \mathrm{~cm},\) calculate the percentage of the metal that is thrown away as scrap.

Short answer

26.98% of the metal is thrown away as scrap without the hole, and 42.96% with the hole.

Step-by-step solution

- Calculate the area of the square

To find out the original amount of metal, calculate the area of the square plate from which the circular part is cut. Use the formula for the area of a square, which is side length squared, i.e., Area = side^2. In this case, the side length of the square is 10.0 cm.

- Calculate the area of the full circle

Next, calculate the area of the full circle that is cut from the square. Use the formula for the area of a circle, which is \(\pi r^2\), where \(r\) is the radius of the circle. The diameter is given as 10.0 cm, so the radius, being half of the diameter, is 5.0 cm.

- Calculate the waste from the full circle

To find the scrap metal percentage without the hole, subtract the area of the circle from the area of the square, and then divide by the area of the square. Multiply the result by 100 to get the percentage. This gives you the percentage of metal wasted when the circle with no hole is cut from the square.

- Calculate the area of the circular hole

Now, you have to calculate the area of the circular hole in the middle of the full circle. Again, use the formula for the area of a circle. The diameter of the hole is 6.0 cm, so its radius is 3.0 cm.

- Calculate the total area of the usable metal part

Subtract the area of the hole from the area of the full circle to get the total area of the usable metal part.

- Calculate the waste including the hole

Finally, to find the scrap metal percentage including the hole, subtract the total area of the usable metal part from the area of the square, and then divide by the area of the square. Multiply by 100 to convert it to a percentage. This will give you the percentage of metal wasted when the circular part with a hole is cut out from the square.

Key concepts

Area of a Square

Calculating the area of a square is one of the fundamental aspects of geometry. This is especially practical when determining the amount of material needed for a project, as in the exercise involving scrap metal calculation. The area of a square is obtained by multiplying the length of one of its sides by itself, which is expressed mathematically as \( \text{Area} = \text{side}^2 \).

For example, if we have a square metal plate with a side length of 10.0 cm, the area of this square will be \( 10.0 \text{cm} \times 10.0 \text{cm} = 100.0 \text{cm}^2 \). Understanding this calculation is crucial, as it represents the total amount of metal we start with before any parts are cut from it.

Area of a Circle

In geometry, the area of a circle is calculated differently from that of a square. The key to finding the area of a circle lies within its radius, which is half the length of the diameter. The formula to find the area is \( \text{Area} = \text{pi} \times \text{radius}^2 \), where pi (\( \text{pi} \)) is approximately 3.14159.

For a circular metal part with a diameter of 10.0 cm, the radius would be half of that, which is 5.0 cm. Thus, the area of the circle is \( \text{pi} \times 5.0^2 \text{cm}^2 \text{ or approximately 78.54 cm}^2 \). To deal with the circular hole in the center, we calculate its area using the same formula. With a diameter of 6.0 cm, the radius is 3.0 cm and the area of the hole is approximately \( 28.27 \text{cm}^2 \). It's important for students to grasp this concept, as it influences the calculation of material that will not be part of the final product and is thus scrap.

Percentage Calculation

Understanding how to compute percentages is essential in many real-world applications, such as analyzing scrap metal in manufacturing processes. A percentage represents a fraction of 100 and is often used to compare sizes of different parts of a whole. To calculate the percentage of metal that becomes scrap after cutting a shape out of a square plate, we follow this general formula: \( \text{Percentage of Scrap} = \frac{\text{Area of Square}-\text{Area of Remaining Shape}}{\text{Area of Square}} \times 100\text{%} \).

The key here is to first determine the total original area and the remaining usable area after the cutting process. For the initial problem without the hole, the percentage of metal that is thrown away is the area of the square minus the area of the circle, divided by the area of the square, all multiplied by 100. When the hole is introduced, you further subtract the area of the hole from the usable area of the circle before performing the percentage calculation. Grasping the concept of percentage calculation is vital as it allows students to quantify waste, an important measure in both environmental and cost efficiency contexts.