Question

Suppose that each dimension of the sheet of paper described in question 1 is increased by one inch. Calculate how many inches the ant will travel in this case. Provide a detailed explanation of your reasoning.

Answer 1

If each dimension of the sheet of paper described in question 1 is increased by one inch, the ant will travel an additional 2 inches.

Step-by-step explanation:

he ant traveled the hypotenuse of a right-angled triangle formed by the dimensions of the sheet of paper. Let the original dimensions be a and b, where a>b. The ant's path, according to the Pythagorean theorem, is given by:

Hypotenuse =
`√( a²+ b²)`

Now, if each dimension is increased by one inch, the new dimensions become a+1 and b+1. The new hypotenuse (H′) is given by:

H′ =
`√((a+1)²+ (b+1)²)`

Subtracting the original hypotenuse from the new hypotenuse gives the additional distance traveled:

Additional Distance=H′ − Hypotenuse

Substituting the expressions for H′ and Hypotenuse:

Additional Distance =
`√((a+1)²+ (b+1)²)` -
`√( a²+ b²)`

For a small increase (1 inch in this case), this can be approximated as:

Additional Distance ≈ 1/2 ((a+1)²+ (b+1)² - a²+ b² /
`√(a²+ b²)`)

Solving for a=8 inches and b=6 inches (as mentioned in question 1):

Additional Distance ≈ 1/2 ((8+1)²+ (6+1)²- (8²+6²) /
`√(8²+6²)`)

Using a calculator to evaluate this expression:

Additional Distance≈2

If each dimension of the sheet of paper is increased by one inch, the ant will travel an additional 2 inches. This is calculated by finding the difference in the hypotenuses of the original and new dimensions using the Pythagorean theorem.