Answer:
1. The number of toppings that would make the costs the same for both pizzas is 5.
2. The side length of the congruent equilateral triangles is approximately 6.67 units.
3. The original length of the rope is 48 inches.
Step-by-step explanation:
For the first problem:
Let \( x \) be the number of toppings. The total cost for Pizza A is \( 10 + 1.5x \), and for Pizza B, it's \( 12.50 + 1x \). To make the costs the same:
\[ 10 + 1.5x = 12.50 + 1x \]
Solve for \( x \):
\[ 1.5x - 1x = 12.50 - 10 \]
\[ 0.5x = 2.50 \]
\[ x = 5 \]
So, 5 toppings would make the costs the same for both pizzas.
For the second problem:
Let \( s \) be the side length of the equilateral triangle. The perimeter of the square is \( 4 + 4 \times 10 = 44 \) units, and the combined perimeter of two congruent equilateral triangles is \( 6s \). So, the equation is:
\[ 44 = 6s + 4 \]
Solve for \( s \):
\[ 6s = 44 - 4 \]
\[ 6s = 40 \]
\[ s = \frac{40}{6} = \frac{20}{3} \]
For the third problem:
Let \( L \) be the original length of the rope. After trimming 8 inches, the remaining length is \( L - 8 \). Each of the 5 equal pieces is \( 40 \) inches less than the original length, so:
\[ \frac{L - 8}{5} = L - 40 \]
Solve for \( L \):
\[ L - 8 = 5(L - 40) \]
\[ L - 8 = 5L - 200 \]
\[ 4L = 192 \]
\[ L = \frac{192}{4} = 48 \]
So, the original length of the rope was 48 inches.
