Problem 1 - Calculus Early Transcendentals

Question

Give an example in which one dimension of a geometric figure changes and produces a corresponding change in the area or volume of the figure.

Step-by-Step Solution

Answer

Question: Provide an example of how changing one dimension of a geometric figure results in a change in its area. Answer: In a rectangle with an initial length of 4 cm and a width of 3 cm, its area is 12 cm^2. If we increase the length to 6 cm, the new area becomes 18 cm^2. By changing the length, the area of the rectangle also changed.

Step 1: Select a geometric figure

Let's choose a rectangle as our geometric figure to keep things simple. A rectangle has two dimensions: length and width.

Step 2: Define the initial dimensions of the rectangle

Let's assume the rectangle has a length of 4 cm and a width of 3 cm. Now let's calculate the area of the rectangle.

Step 3: Calculate the initial area of the rectangle

The area of a rectangle can be found using the formula A = length × width. In our case, the area of the rectangle is: A = 4 cm × 3 cm = 12 cm^2

Step 4: Modify one dimension of the rectangle

Now, let's increase the length of the rectangle by 2 cm, making the new length 6 cm. The width will remain the same at 3 cm.

Step 5: Calculate the new area of the rectangle

With the new dimensions, we can calculate the rectangle's new area using the same formula mentioned in Step 3: A_new = 6 cm × 3 cm = 18 cm^2

Step 6: Compare the initial and new areas

Let's now compare the initial area and the new area of the rectangle: Initial area: 12 cm^2 New area: 18 cm^2 We can see that by changing one dimension of the rectangle, the length, its area has also changed accordingly. In this example, we increased the length by 50% (from 4 cm to 6 cm), resulting in a 50% increase in the area (from 12 cm^2 to 18 cm^2).