$4$ . CRAFTS Marina crafts unique trivets and other kitchenware. Her basic trivet design is an equilateral triangle with an area of about $3$ . $9$ square inches. She plans to make trivet $A$ by increasing each side by $\frac{4}{3}$ . To make trivet $B$ , Marina will double the length of one side while keeping the height as measured from the doubled side the same as the basic trivet. What are the approximate areas of trivets $A$ and $B$ ?

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4

. CRAFTS Marina crafts unique trivets and other kitchenware. Her basic trivet design is an equilateral triangle with an area of about

3

9

square inches. She plans to make trivet

A

by increasing each side by

\frac{4}{3}

. To make trivet

B

, Marina will double the length of one side while keeping the height as measured from the doubled side the same as the basic trivet. What are the approximate areas of trivets

A

and

B

Trivet A Side Length Increase: For trivet A, each side is increased by $(4)/(3)$ . The area of an equilateral triangle is given by the formula $A = (\sqrt{3}/4) \times s^2$ . Let's call the original side length $'s'$ . The new side length for trivet A is $s \times (4/3)$ .
Calculate New Area for Trivet A: Now, calculate the new area for trivet A using the new side length $\frac{4s}{3}$ . Area of trivet A = $\frac{\sqrt{3}}{4} \times \left(\frac{4s}{3}\right)^2$ .
Simplify Area Formula for Trivet A: Simplify the area formula for trivet A: Area = $(\sqrt{3}/4) \times (16s^2/9) = (4\sqrt{3}/4) \times (s^2/9) = (\sqrt{3}/9) \times s^2$ .
Calculate $s^2$ for Trivet A: Since the original area is $3.9$ square inches, we can set $(\sqrt{3}/4) * s^2 = 3.9$ and solve for $s^2$ . $s^2 = 3.9 / (\sqrt{3}/4) = 3.9 * (4/\sqrt{3}) = 3.9 * (4\sqrt{3}/3)$ .
Plug $s^2$ into Area Formula for Trivet A: Calculate $s^2$ : $s^2 = 3.9 \times \left(\frac{4\sqrt{3}}{3}\right) = \frac{15.6\sqrt{3}}{3}$ .
Simplify Area for Trivet A: Now, plug $s^2$ into the area formula for trivet A: Area of trivet A = $(\sqrt{3}/9) \times 15.6\sqrt{3}/3$ .
Trivet B Side Length Doubling: Simplify the area for trivet A: Area = $\frac{\sqrt{3} \times 15.6\sqrt{3}}{9 \times 3}$ = $\frac{15.6 \times 3}{9}$ = $5.2$ square inches.
Calculate New Area for Trivet B: For trivet B, double the length of one side but keep the height the same. The area of a triangle is also given by $A = \frac{1}{2} \times \text{base} \times \text{height}$ . If we double the base and keep the height the same, the new area will be double the original area.
Calculate New Area for Trivet B: For trivet B, double the length of one side but keep the height the same. The area of a triangle is also given by $A = \frac{1}{2} \times \text{base} \times \text{height}$ . If we double the base and keep the height the same, the new area will be double the original area.Calculate the new area for trivet B: Area of trivet B = $2 \times \text{original area} = 2 \times 3.9 = 7.8$ square inches.