Q. A rectangular field is 50 yards wide and 100 yards long. Patrick walks diagonally across the
field. How far does he walk?
answer choices
111.8 yards
127.3 yards
115 yards
119 yards
( plz show work

111.8 yards

Step-by-step explanation:

So by walking diagonally across the field, Patrick is making a right triangle with his path (the two legs are the width and length of the field). Let a = 50, b = 100, and c = the distance Patrick walks.

For a right triangle, we use the Pythagorean Theorem: a² + b² = c²

Since you're finding c, we can rearrange (using algebra) to get:

Fill in the values for a and b: = 111.8 yards.

Since we want to solve for the variable x, we want to isolate x

a²x + (a - 1) = (a + 1)x ⇒ Distribute x to (a+1). Also, remove parentheses

a²x + a - 1 = ax + x ⇒ Subtract a from both sides

a²x - 1 = ax + x - a ⇒ Add 1 to both sides

a²x = ax + x - a + 1 ⇒ Subtract (ax + x) from both sides

a²x - (ax + x)= ax + x - a + 1 - (ax+x) ⇒ Simplify. Remember that multiplying positive by negative = negative

a²x - ax - x = ax + x - a + 1 - ax - x ⇒ Simplify

a²x - ax - x = -a + 1 ⇒ Factor out the x from a²x - ax - x

x(a² - a - 1) = -a + 1 ⇒ Divide both sides by (a² - a - 1)

x = (-a + 1) / (a² - a - 1)

However, we need to make sure that the denominator does not equal 0. Therefore, you set the denominator = 0 (just use the quadratic formula for this), and it gives that the denominator =0 when a = (1+√5)/2 AND (1-√5)/2

Therefore, the final answer is

x = (-a + 1) / (a² - a - 1) given that a ≠ (1+√5)/2, a ≠ (1-√5)/2