Final answer:
The lateral surface areas of the red and yellow umbrellas are both 30 square feet, while the blue umbrella's lateral surface area is slightly larger at 30.4 square feet. Given the small difference, your friend can choose her favorite color without significant impact on coverage.
Step-by-step explanation:
To calculate the lateral surface area of a pyramid, we first need to find the area of one triangular face and then multiply it by the number of faces. The area of one triangle is \(\frac{1}{2} \times \text{{side length}} \times \text{{slant height}}\). For the regular pentagonal pyramid, there are five faces; for the hexagonal pyramid, there are six; and for the octagonal pyramid, there are eight faces.
a. For the red umbrella (pentagonal pyramid) with a side length of 3 feet and a slant height of 4 feet, the area of one triangle is \(\frac{1}{2} \times 3 \times 4 = 6 \text{{ square feet}}\). Therefore, the lateral surface area is \(6 \times 5 = 30 \text{{ square feet}}\).
b. For the yellow umbrella (hexagonal pyramid) with a side length of 2.5 feet and a slant height of 4 feet, the area of one triangle is \(\frac{1}{2} \times 2.5 \times 4 = 5 \text{{ square feet}}\). Thus, the lateral surface area is \(5 \times 6 = 30 \text{{ square feet}}\).
c. For the blue umbrella (octagonal pyramid) with a side length of 1.9 feet and a slant height of 4 feet, the area of one triangle is \(\frac{1}{2} \times 1.9 \times 4 = 3.8 \text{{ square feet}}\). Then, the lateral surface area is \(3.8 \times 8 = 30.4 \text{{ square feet}}\).
d. Based on lateral surface areas, you can suggest that your friend pick the umbrella in her favorite color, as the difference in surface areas is minimal. All three umbrellas have similar surface areas, with the red and yellow umbrellas having exactly the same area, and blue having a slightly larger area. The choice can therefore be based on color preference without much sacrifice in terms of protection from the elements.