Jack's Responses in WyzAnt Answers

Assuming that the house is perpenidcular to the ground on which the ladder stands, use the Pythagorean Theorem for the sides of right triangles:

           a² + b²  =  c²

In this problem a = 18, b is the unknown, and c is the length of the ladder (the hypotenuse of the triangle).

          18² + b²  =  22²   or

                     b²  =  22² - 18²

                     b²  =  484 - 324

                     b²  =  160

                     b    =  √160  =  12.6 ft.

(It would be hard to climb that ladder because it is farther away from the wall than the height of the place where it touches the house)

Hello, Jesse --

You need to use your information about area and perimeter to set up two equations with two variables representing length and width of the rectangle. Then you will set up a quadratic equation. Let's use L for the length and W for the width of the rectangle:

For the perimeter:

          2L + 2W = 66  or  
             L + W  = 33  or
                   W  = 33 - L

For the area:

             LW =  252

Now substitute W = 33 - L from the first equation into the second one:

             L(33 - L) = 252  or

              33L - L² = 252  or

               L² - 33L + 252 = 0

This can be solved by factoring, or by using the quadratic equation.

By factoring:

          (L - 12)(L - 21) = 0  or

                  L = 12, or L = 21

In this case, let's use 21 because length is usually the longer dimension. Now we substitute L = 21 into the first equation to find W.

        W = 33 - 21 or
         W = 12

Your answers are:  Length = 21 and Width = 12

Gina's answer is correct.

We can also set it up as a ratio:

      (Original number of packs)/Original cost of those packs)  =  (New number of packs)/(New cost)

In this problem, that would be:

           3/0.87  =  10/x     (where x is the cost of 10 packs)

Cross-multiplying, we get:

     3x  =  8.70

        x  =  2.90

With instructions to use exponents, I would do it this way:

2n²(8)³  = 2(2³)³ n² = 2(2⁹) n² = 2¹⁰ n²

This is exactly the same as Tamara's answer above, but with an explicit powr of 2.

Hello, Diana --

This will involve a u-substitution and a factored quadratic.

Let u = cos(ln(3x))

Then 2u² + u - 1 = 0

Factor:     (2u - 1)(u + 1) = 0

        2u - 1  =  0    or   u + 1  = 0

          u   =  1/2           u  =  -1

Resubstitute:

       cos(ln(3x))  = 1/2      cos(ln(3x)) = -1

         ln(3x)  =  pi/3                 ln(3x)  = pi

   ln 3 + ln x   = pi/3          ln 3 + ln x  = pi

         ln x = pi/3 - ln 3              ln x =  pi - ln 3

           x  =  e^{(pi/3 - ln 3)}              x  = e^{(pi - ln 3)}

There may be other solutions that are integer multiples of the pi/3 and pi compnents.