1. The total number of lights in a triangular lighting rig is related to the triangular numbers, as shown in the diagram. The sum of the first n triangular numbers is given by the polynomial function shown below. Find the sum of the first seven triangular numbers.

D. 84

2. Given the value of g(7) from the previous question, we can write the equation seen here. (Multiply both sides by six and subtract the constant term from both sides.) Factor this equation. (Hint: You know that the equation is true for n = 7.)

D. (n - 7)(n2 + 10n + 72)

3. The total number of lights in a triangular rig with n rows is given by the function T(n) as shown here. Find the number of lights in triangular rigs with 1 to 10 rows.

C. 1, 3, 6, 10, 15, 21, 28, 36, 45, 55

4.T(n) is the total number of lights in a triangular lighting rig of n rows, as shown. Which polynomial function represents T(n + 1)?

B. T(n+1)= 1/2n^2 + 3/2n + 1

5. A square lighting rig's total number of lights can be calculated with the equation S(n) = n2, where n is the number of rows. In stadiums and other large venues, it may be necessary to group lighting rigs together into giant arrays of rigs on steel frameworks. If triangular lighting rigs are grouped together into a giant square array, the total number of lights would be given by multiplying the triangular rig function by the square rig function. Assuming that the number of rows n in the triangular rigs is the same as the number of rows n in the overall square array, what polynomial expression allows us to calculate the total number of individual lights in this super lighting rig?

A. 1/2n^3(n+1)

6. The number of lights in a pentagonal rig with n rows is given by the function shown below. Find the number of lights in pentagonal rigs with 1 to 10 rows.

B. 1, 5, 12, 35, 51, 70, 92, 117, 145

7. Which of these polynomial functions gives the number of lights in a hexagonal rig (six sides) with n rows? (Hint: Look for a pattern in the coefficients of the polynomial functions for the triangular, square, and pentagonal lighting rigs.)

A. H(n) = 2n^2 - n
B. H(n) = 2n^2 - 1
C. H(n) = ½n^2 + n + 1
D. H(n) = ½n^2 + n

8. H(n) ÷ S(n) will give the ratio between the number of lights in a hexagonal rig and a square rig. After doing the division, determine which one of these statements is true.

A. As the number of rows increases, the hexagonal rig gets closer to having three times as many lights as the square rig.
B. As the number of rows decreases, the hexagonal rig gets closer to having twice as many lights as the square rig.
C. As the number of rows decreases the hexagonal rig gets closer to having three as many lights as the square rig.
D. As the number of rows increases, the hexagonal rig gets closer to having twice as many lights as the square rig.

9. The total number of lights in an octagonal lighting rig (with eight sides) with n rows can be found with the function O(n) = 3n2 - 2n. Which of these shows both the normal factorization of this polynomial and the factorization that reveals the general pattern for the number of lights in a rig with any number of sides?

C. n(3n - 2) and ½n(6n - 4)

10. A lighting manufacturer calculates the cost of producing triangular lighting rigs with the function C(x) = 2x3 + 7x2 + 5x, where x is the number of triangular rigs in the order. Which of these represents the factored version of this polynomial?

B. x (x + 1)(2x + 5)

11. The lighting manufacturer uses sheet metal hoods to reflect the light towards the stage. They define the shape of the sheet metal with a polynomial in two variables: 3x3 - 81y3. Factor this polynomial.

D. 3(x - 3y)(x^2 - 3xy + 9y^2)

Question

1. The total number of lights in a triangular lighting rig is related to the triangular numbers, as shown in the diagram. The sum of the first n triangular numbers is given by the polynomial function shown below. Find the sum of the first seven triangular numbers.

D. 84

2.  Given the value of g(7) from the previous question, we can write the equation seen here. (Multiply both sides by six and subtract the constant term from both sides.) Factor this equation. (Hint: You know that the equation is true for n = 7.)

D. (n - 7)(n2 + 10n + 72)

3.  The total number of lights in a triangular rig with n rows is given by the function T(n) as shown here. Find the number of lights in triangular rigs with 1 to 10 rows.

C. 1, 3, 6, 10, 15, 21, 28, 36, 45, 55

4.T(n) is the total number of lights in a triangular lighting rig of n rows, as shown. Which polynomial function represents T(n + 1)?

B. T(n+1)= 1/2n^2 + 3/2n + 1

5. A square lighting rig's total number of lights can be calculated with the equation S(n) = n2, where n is the number of rows. In stadiums and other large venues, it may be necessary to group lighting rigs together into giant arrays of rigs on steel frameworks. If triangular lighting rigs are grouped together into a giant square array, the total number of lights would be given by multiplying the triangular rig function by the square rig function. Assuming that the number of rows n in the triangular rigs is the same as the number of rows n in the overall square array, what polynomial expression allows us to calculate the total number of individual lights in this super lighting rig?

A. 1/2n^3(n+1)

6. The number of lights in a pentagonal rig with n rows is given by the function shown below. Find the number of lights in pentagonal rigs with 1 to 10 rows.

B. 1, 5, 12, 35, 51, 70, 92, 117, 145

7. Which of these polynomial functions gives the number of lights in a hexagonal rig (six sides) with n rows? (Hint: Look for a pattern in the coefficients of the polynomial functions for the triangular, square, and pentagonal lighting rigs.)

A. H(n) = 2n^2 - n
B. H(n) = 2n^2 - 1
C. H(n) = ½n^2 + n + 1
D. H(n) = ½n^2 + n

8. H(n) ÷ S(n) will give the ratio between the number of lights in a hexagonal rig and a square rig. After doing the division, determine which one of these statements is true.

A. As the number of rows increases, the hexagonal rig gets closer to having three times as many lights as the square rig.
B. As the number of rows decreases, the hexagonal rig gets closer to having twice as many lights as the square rig.
C. As the number of rows decreases the hexagonal rig gets closer to having three as many lights as the square rig.
D. As the number of rows increases, the hexagonal rig gets closer to having twice as many lights as the square rig.

9. The total number of lights in an octagonal lighting rig (with eight sides) with n rows can be found with the function O(n) = 3n2 - 2n. Which of these shows both the normal factorization of this polynomial and the factorization that reveals the general pattern for the number of lights in a rig with any number of sides?

C. n(3n - 2)  and  ½n(6n - 4)

10. A lighting manufacturer calculates the cost of producing triangular lighting rigs with the function C(x) = 2x3 + 7x2 + 5x, where x is the number of triangular rigs in the order. Which of these represents the factored version of this polynomial?

B. x (x + 1)(2x + 5)

11. The lighting manufacturer uses sheet metal hoods to reflect the light towards the stage. They define the shape of the sheet metal with a polynomial in two variables: 3x3 - 81y3. Factor this polynomial.

D. 3(x - 3y)(x^2 - 3xy + 9y^2)

anonymous · Answer

I think we need the diagram.

alyssajobug · Answer

attachment

alyssajobug · Answer

2.

alyssajobug · Answer

3.

alyssajobug · Answer

4

alyssajobug · Answer

5.

anonymous · Answer

For 1 and 3, all you do is plug in the correct value into the function they give you.
For 1, that value is 7 since they asked for the total number of (blah blah blah) for the first seven.
For 3, you are to make a list of results you got by plugging in every whole number from 1 to 10 and seeing which answer it matched with.

alyssajobug · Answer

6.

anonymous · Answer

For 2, since you already know that the equation is true for n=7, it is confirmed that n-7 is a factor of the function. (recall that the point of finding factors of a polynomial function is to find all values for which T(n)=0 holds true.) Since you are already given n-7, you can go aha and factor that out from The function by long division. What remains should be a quadratic function that should be manageable to factor.

anonymous · Answer

For 4, replace n with (n+1) (don't forget the brackets), expand, and combine like terms to get it in the form of a(n^2)+b(n)+c (where a,b,and c are all numerical constants)

anonymous · Answer

And holy crap this question is long.

anonymous · Answer

For 5, do as the question tells you and multiply T(n) from part 3 to S(n) given in this part.

anonymous · Answer

6 works exactly the same as 3

anonymous · Answer

For 7, observe that when you went from triangular to square, the function had ((n^2)/2)-(n/2) add to it, and the same thing occurred from square to pentagonal. Thus, by this logic, the transformation from pentagonal to hexagonal must also undo this exact same transformation.
Thus H(n)=P(n)+((n^2)/2)-(n/2)=2(n^2)-n

anonymous · Answer

For 8, the question is asking you to try dividing H(n) by S(n) and observing the ratio.
H(n)/S(n)=2-(1/n)
1/n approaches 0 as the value of n gets larger and larger, so as m gets larger, H(n) gets closer and closer to being twice as large as S(n)

anonymous · Answer

For 9, factor n out of the equation they gave you to get the factorization for O(n). For the general case, I have no idea what it's trying to k, so can't help you there.

anonymous · Answer

Trying to ask**

anonymous · Answer

For 10, you are asked to factor that...thing they gave you in the question. Notice that you can immediately pull out x from the equation, which turns the remaining into a quadratic function easy enough to manage.

anonymous · Answer

11 is too much work for me to even attempt to figure out, but the general idea is to add 0xy^2 + 0 yx^2 into there and attempt to factor it after that.

anonymous · Answer

If you ever get through this, you have the right to complain to your teacher about this attempt to heinously murder you with all this tedious work. Although you still don't have as much of a right as university students and their calculus.

alyssajobug · Answer

I have to take calculus in HS cuz I'm taking this as a sophomore :-/ AP will kill me.