4. You are at Lowes and a man is leaning against three sections of fencing that are 6,7 and 12 ft long. A Lowe’s person tells him to discard the 12 ft section and buy a 10ft section. She informs the customer that this would save money and would fence a bigger triangular area for his dog. At first you think she is wrong but later you determine she is right. How much more area does the dog gain if the owner gets the 10 ft section? Round to the nearest 10th of a square foot.

To determine how much more area the dog gains by using the 10 ft section instead of the 12 ft section, we need to calculate the area of the two possible triangular enclosures.

For the initial scenario with the 12 ft section, we can use Heron's formula to calculate the area. The formula is as follows:

Area = √(s(s-a)(s-b)(s-c))

where s is the semi-perimeter (half the perimeter) of the triangle, and a, b, and c are the lengths of the sides. In this case, the sides are 6 ft, 7 ft, and 12 ft.

Calculating the semi-perimeter:

s = (6 + 7 + 12) / 2 = 12.5 ft

Calculating the area:

Area_initial = √(12.5(12.5-6)(12.5-7)(12.5-12))
Area_initial ≈ 20.44 square feet

Now, let's calculate the area when the 10 ft section is used instead:

The sides of the triangle would now be 6 ft, 7 ft, and 10 ft.

Calculating the semi-perimeter:

s = (6 + 7 + 10) / 2 = 11.5 ft

Calculating the area:

Area_final = √(11.5(11.5-6)(11.5-7)(11.5-10))
Area_final ≈ 20.99 square feet

To find how much more area the dog gains, subtract the initial area from the final area:

Area_gain = Area_final - Area_initial
Area_gain ≈ 20.99 - 20.44
Area_gain ≈ 0.55 square feet

Therefore, by using the 10 ft section instead of the 12 ft section, the dog gains approximately 0.55 square feet of extra area.

To determine the increase in area gained by the dog if the owner gets the 10 ft section instead of the 12 ft section, we can use the formula for the area of a triangle, which is given by the formula:

Area = 1/2 * base * height

In this case, the three sections of fencing will be the sides of the triangular dog area. Let's label them as follows:

a = 6 ft
b = 7 ft
c = 12 ft

To find the area of the original triangular area, we can use Heron's formula, which calculates the area of a triangle given its side lengths:

Area_original = sqrt(s * (s - a) * (s - b) * (s - c))

where s is the semi-perimeter of the triangle. The semi-perimeter is calculated by adding up the lengths of all three sides and dividing by 2:

s = (a + b + c) / 2

Once we have the area of the original triangular area, we can calculate the area of the new triangular area with the 10 ft section by substituting the side lengths as follows:

a = 6 ft
b = 7 ft
c = 10 ft

We can then use the same formula to calculate the new area.

Finally, to determine the increase in area gained by the dog, we subtract the area of the original triangle from the area of the new triangle.

Let's calculate the area gained step by step:

1. Calculate the semi-perimeter of the original triangle:
s_original = (a + b + c) / 2 = (6 + 7 + 12) / 2 = 12.5 ft

2. Calculate the area of the original triangle:
Area_original = sqrt(s_original * (s_original - a) * (s_original - b) * (s_original - c))
Area_original = sqrt(12.5 * (12.5 - 6) * (12.5 - 7) * (12.5 - 12))
Area_original ≈ 15.92 ft²

3. Calculate the semi-perimeter of the new triangle:
s_new = (a + b + c_new) / 2 = (6 + 7 + 10) / 2 = 11.5 ft

4. Calculate the area of the new triangle:
Area_new = sqrt(s_new * (s_new - a) * (s_new - b) * (s_new - c_new))
Area_new = sqrt(11.5 * (11.5 - 6) * (11.5 - 7) * (11.5 - 10))
Area_new ≈ 15.97 ft²

5. Calculate the increase in area gained by the dog:
Increase in area = Area_new - Area_original
Increase in area ≈ 15.97 ft² - 15.92 ft² ≈ 0.05 ft²

Therefore, the dog gains approximately 0.05 square feet of additional area by using the 10 ft section instead of the 12 ft section.