Tamara's Responses in WyzAnt Answers

Recall the standard form for the equation of a hyperbola is as follows:

     Horizontal transverse axis:     (x - h)²/a²  -  (y - k)²/b²  = 1  

     Vertical transverse axis:     (y - k)²/a²  -  (x - h)²/b²  = 1 

The equation of the hyperbola in question is already in standard form:

          (y - 3)²/16  -  (x + 2)²/9  = 1

Looking at the equation, we see that the transverse axis is vertical. With this, we can find the center, vertices, foci, and asymptotes.

   h = -2 ,   k = 3      ==>     center:   (h, k) = (-2, 3)

   a² = 16     ==>     √a² = √16     ==>     a = 4

   b² = 9     ==>     √b² = √9     ==>     b = 3

   to find c, use the following formula:      c² = a² + b² 

                                                            c² = 16 + 9

                                                            c² = 25      ==>      c = 5

Vertices:     (h, k + a)  =  (-2, 3 + 4) = (-2, 7)

                    (h, k - a)  =  (-2, 3 - 4) = (-2, -1)

Foci:     (h, k + c)  =  (-2, 3 + 5) = (-2, 8)

             (h, k - c)  =  (-2, 3 - 5) = (-2, -2)

Asymptotes:     y - k = ± (a/b)(x - h)

                         y - 3 = ± (4/3)(x + 2)

         thus, the asymptotes of this hyperbola have the following equations:

            y - 3 = (4/3)(x + 2)          and          y - 3 = -(4/3)(x + 2)

When multiplying three or more real numbers, their product is always the same regardless of their grouping.

For instance, let  a, b, and c  be real numbers then, by the above statement, the following is true:

          (a • b) • c  =  a • (b • c)  =  (a • c) • b

This is known as the associative property of multiplication.

Thus,

          9 • (6 • n)  =  (9 • 6) • n  =  (9 • n) • 6

Given:

     Right triangle A:     base = b     ,      height = h     , and     Area = x

     Recall that the area (A) of a triangle is given be the following formula:  

                                 A = (1/2)bh

     Thus, since we are given that the area of triangle A is equal to x, then

                                (1/2)bh = x

     Rectangle B:     length = l = 2b     and      width = w = 2h

     Recall that the area (A) of a rectangle is given by the following formula:  

                                             A = lw

     Thus, the area (A) of rectangle be is as follows:

                                          A = 2b·2h

                                          A = 2·2(bh)

                                          A = 4(bh)

From the area of triangle A, we can solve for bh then substitute this expression in terms of x into the formula of the area of rectangle B above:

    area of triangle A:     (1/2)bh = x      multiply both sides of equation by 2 to solve for bh

                                   2·(1/2)bh = 2·x

                                   bh = 2x

   area of rectangle B:     A = 4(bh)     given bh=2x, replace bh in this equation with 2x

                                     A = 4(2x)

                                     A = 8x

Thus, the area of rectangle B in terms of x is 8x

Recall:     Profit  =  Revenue  -  Cost

                 P(x)  =  R(x)  -  C(x)   ,     where x is the number of pizzas sold

   Revenue, R(x), is given by the following:

     R(x) = (selling price)·(# of pizzas sold) 

     R(x) = 12x

   Cost, C(x), is given by the following:

     C(x) = (variable cost)·(# of pizzas sold) + fixed cost

     C(x) = 5x + 230

Therefore, since    Profit = Revenue - Cost

          P(x) = R(x) - C(x)

           P(x) = 12x - (5x + 230)

           P(x) = 12x - 5x - 230

          P(x) = 7x - 230

Thus, the profit of selling 85 pizzas is determined by solving P(x) when x=85:

          P(x) = 7x - 230

      P(85) = 7(85) - 230

                = 595 - 230

                = 365

Since   P(85) = 365 ,  selling 85 pizzas yield a profit of $365.

To find how many pizzas need to be sold to yield a profit of $700, solve for x when P(x)=700:

                  P(x) = 7x - 230

                 700 = 7x - 230

        700 + 230 = 7x - 230 + 230

                 930 = 7x

              930/7 = 7x/7

           132.86 = x

                    x ≈ 133

Thus, to make a profit of $700, 133 pizzas need to be sold.

To find how many pizzas need to be sold to yield a profit greater than $1000, solve for x when P(x)>1000:

          P(x) = 7x - 230

                  7x - 230 > 1000

          7x - 230 + 230 > 1000 + 230

                            7x > 1230

                         7x/7 > 1230/7

                              x > 175.7

                              x ≥ 176

Thus, to make a profit greater than $1000, you would need to sell at least 176 pizzas.

The area (A) of a triangle given its base (b) and height (h) is defined by the following formula:

          Area = (1/2)base·height     ==>     A = (1/2)bh

The triangle in question has the following base and height:

               b = 7 cm      and      h = 13 cm

Therefore, its area is:

                    A = (1/2)(7 cm)(13 cm)

                       = (1/2)(91 cm²)

                   A = 45.5 cm²