Hi! Can someone help me get started with these 6 problems, please? Would be a huge help! :D
1. A biologist studied the populations of common guppies and Endler’s guppies over a 6-year period. The biologist modeled the populations, in tens of thousands, with the following polynomials where x is time, in years. Common guppies: 3.22 + 5 + 0.2 Endler’s guppies: 4.42 − 5.1 + 1 What polynomial models the total number of common and Endler’s guppies?
2. A family is building a circular fountain in the backyard. The yard is rectangular and measures 10x by 15x and the fountain is going to be circular with a radius of 4x. Once the fountain is built, what will be the area of the remaining yard?
3. A sports team is building a new stadium on a rectangular lot of land. The lot measures 8x by 12x and the field will be 3x by 6x. How much land will be left for the bleachers, restrooms, and other parts of the stadium?
4. A cylinder has a radius of + 2 and a height of + 4. Which polynomial in standard form best describes the total volume of the cylinder? Use the formula = 2ℎ for the volume of a cylinder.
5. A sphere has a radius of 2 + 3. Write the polynomial in standard form that best describes the total surface area of the sphere? Use the formula = 42 for surface area of a sphere.
6. A carpenter is putting a skylight in a roof. If the roof measures (10x + 9) by (7x + 7) and the skylight measures (x + 5) by (3x + 3), what is the area of the remaining roof after the skylight is built?
#1. which "following polynomials"?
Check each to see which contains the given points.
#2. the difference in the areas is length*width - πr^2
#3. very similar to #2, only both areas are just rectangles, right?
#4. Sounds like you are missing some x's. I'd guess that
radius = x+2
height = x+4
So, v = πr^2 h = π(x+2)^2 (x+4)
#5. Again, your copy/paste have produced garbage. But the area of a sphere is just
a = 4πr^2
so plug in your correct value for r.
#6. just like #3.
Sure, I'd be happy to help you get started with these problems! Let's break down each problem and explain how to find the solution.
1. To find the polynomial that models the total number of common and Endler's guppies, you simply need to add the polynomial for the common guppies to the polynomial for the Endler's guppies.
The polynomial for common guppies is 3.22 + 5 + 0.2x, and the polynomial for Endler's guppies is 4.42 − 5.1 + x. Adding these two polynomials together gives you:
Total number of guppies = (3.22 + 5 + 0.2x) + (4.42 − 5.1 + x).
2. To find the area of the remaining yard after the circular fountain is built, you need to calculate the area of the entire yard and subtract the area of the circular fountain.
The yard is rectangular and measures 10x by 15x, so its area is given by the formula Area = length × width. In this case, the length is 15x and the width is 10x, so the area of the yard is (15x)(10x) = 150x^2.
The circular fountain has a radius of 4x, so its area is given by the formula Area = πr^2, where r is the radius. Substituting the value of the radius, we get the area of the fountain as π(4x)^2 = 16πx^2.
To find the area of the remaining yard, subtract the area of the circular fountain from the area of the entire yard: Remaining area = 150x^2 - 16πx^2.
3. To find the land left for the bleachers, restrooms, and other parts of the stadium, you need to subtract the area of the field from the area of the entire lot.
The lot is rectangular and measures 8x by 12x, so its area is given by the formula Area = length × width. In this case, the length is 12x and the width is 8x, so the area of the lot is (12x)(8x) = 96x^2.
The field is also rectangular and measures 3x by 6x, so its area is (3x)(6x) = 18x^2.
To find the land left for the bleachers, restrooms, and other parts of the stadium, subtract the area of the field from the area of the entire lot: Land left = 96x^2 - 18x^2.
4. To find the polynomial that best describes the total volume of the cylinder, you need to use the formula for the volume of a cylinder, which is V = πr^2h.
The cylinder has a radius of 2 and a height of 4, so substituting these values into the formula, we get: V = π(2)^2(4) = 16π.
5. To find the polynomial that best describes the total surface area of the sphere, you need to use the formula for the surface area of a sphere, which is A = 4πr^2.
The sphere has a radius of 2 + 3, so substituting this value into the formula, we get: A = 4π(2 + 3)^2 = 4π(25) = 100π.
6. To find the area of the remaining roof after the skylight is built, you need to calculate the area of the entire roof and subtract the area of the skylight.
The roof is rectangular and measures (10x + 9) by (7x + 7), so its area is given by the formula Area = length × width. In this case, the length is (7x + 7) and the width is (10x + 9), so the area of the roof is (7x + 7)(10x + 9).
The skylight is also rectangular and measures (x + 5) by (3x + 3), so its area is given by (x + 5)(3x + 3).
To find the area of the remaining roof, subtract the area of the skylight from the area of the entire roof: Remaining area = (7x + 7)(10x + 9) - (x + 5)(3x + 3).
I hope this explanation helps you get started with these problems! Let me know if you have any further questions.