Question

Multiple questions that I have no idea how to fix!!!


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.

A. 77
B. 67
C. 74
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.)

A. (n - 21)(n2 + 10n + 24)
B. (n - 21)(n + 6)(n + 4)
C. (n - 7)(n + 6)(n + 4)
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.

A. 1, 3, 6, 10, 13, 16, 20, 23, 26, 30
B. 1, 3, 6, 10, 15, 20, 25, 30, 35, 40
C. 1, 3, 6, 10, 15, 21, 28, 36, 45, 55
D. 1, 3, 6, 10, 15, 22, 29, 37, 46, 56

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)?

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?

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.

A. 1, 5, 12, 30, 47, 59, 76, 91, 118
B. 1, 5, 12, 35, 51, 70, 92, 117, 145
C. 1, 5, 12, 35, 53, 77, 98, 129, 164
D. 1, 5, 12, 30, 49, 63, 85, 103, 125

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?

A. n(2n - 3) and ½n(6n - 4)
B. n(3n - 2) and ½n(6n - 4)
C. n(3n - 2) and ½n(1.5n - 1)
D. n(2n - 3) and ½n(1.5n - 1)

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?

A. x^2(x + 1)(2x + 5)
B. x (x + 1)(2x + 5)
C. 2x (x + 1)(x + 5)
D. 2x (x + 1)^2 (x + 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.

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