Final answer:
The total guppy population is modeled by the polynomial 7.64x^2 - 0.1x + 1.2. For remaining areas, we subtract the area of the structure (fountain, field, skylight) from the total area of the yard, lot, or roof, using appropriate equations for area calculation.
The correct answers are D, A, A, B, A, and A respectively.
Step-by-step explanation:
To model the total number of common and Endler’s guppies, we simply add the individual polynomials for each guppy population.
Common guppies: (3.22x^2 + 5x + 0.2)
Endler’s guppies: (4.42x^2 - 5.1x + 1)
Total guppies: (3.22x^2 + 5x + 0.2) + (4.42x^2 - 5.1x + 1) = (3.22x^2 + 4.42x^2) + (5x - 5.1x) + (0.2 + 1) = 7.64x^2 - 0.1x + 1.2, so the answer is d).
To find the area of the remaining yard after building a circular fountain, we calculate the area of the yard and then subtract the area of the fountain. The area of the yard is the product of its length and width, 10x by 15x, so 150x^2. The area of the fountain is the area of a circle with radius 4x, which is π(4x)^2 or 16πx^2. Subtracting the fountain area from the yard area gives 150x^2 - 16πx^2, so answer a) is correct.
The land remaining for other parts of the stadium is calculated by subtracting the area of the field from the total lot area. The lot area is 8x × 12x = 96x^2. The field area is 3x × 6x = 18x^2. The difference is 96x^2 - 18x^2, which gives us 78x^2, so answer a) is correct.
The total volume of the cylinder with radius 2 and height 4 is given by the formula V = πr^2h, which results in π(2)^2(4) = 16π, thus the answer is b).
The surface area of a sphere with radius 2 is 4πr^2, giving us 4π(2)^2 = 16π, so the answer is a).
The area of the remaining roof after the skylight is built is found by subtracting the area of the skylight from the area of the roof, which gives (10x + 9)(7x + 7) - (x + 5)(3x + 3), correspondingly the answer is a).