Geometry

1. The triangle inequality theorem states that the sum of any two sides must be greater than the third side. Check all three conditions:

• $5 + 8 = 13$ and $13 < 14$ ✗

Since $5 + 8 < 14$, the condition fails. No, a triangle cannot be formed with these side lengths.

2. Volume:

$V = l \times w \times h = 10 \times 4 \times 3 = 120 \text{ in}^3$

Surface area (sum of all six faces):

$SA = 2(lw + lh + wh) = 2(10 \times 4 + 10 \times 3 + 4 \times 3)$ $SA = 2(40 + 30 + 12) = 2(82) = 164 \text{ in}^2$

3. Since the included angle is $90°$, this is a right triangle with legs 6 cm and 9 cm.

$A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 9 = 27 \text{ cm}^2$

4. The volume of a cylinder is $V = \pi r^2 h$:

$V = 3.14 \times 3^2 \times 7 = 3.14 \times 9 \times 7 = 3.14 \times 63 = 197.82 \text{ ft}^3$

The volume of the tank is approximately $197.82 \text{ ft}^3$.

5. First find the area of the triangular base:

$A_{\text{base}} = \frac{1}{2} \times 6 \times 4 = 12 \text{ m}^2$

The volume of a prism is base area times length:

$V = A_{\text{base}} \times \text{length} = 12 \times 10 = 120 \text{ m}^3$