a. Colin purchased an extension ladder consisting of two 7-foot
sections. When fully extended, the ladder measures 11 feet 9 inches.
By how much do the two ladder sections overlap?
b. Find the area of a regular pentagon with sides 7 mm long and
apothem 4.8 mm long (A = 1/2aP)
c. Solids A and B are similar. The height of solid A is 6 meters and
the height of solid B is 15 meters. If the volume of solid B is 250
cubic meters, what is the volume of solid a?
d. A triangular prism has a height of 12 m. What is the surface area
of the prism if its bases are right triangles with legs 3 meters and 4
meters?
  (S = Ph + 2B)
e. Use the Law of Detachment or the Law of Syllogism to determine a
conclusion that follows from statements (1) and (2). If a valid
conclusion does not follow, write no valid conclusion.
   (1) ALL SQUARES ARE RECTANGLES.
   (2) IF A QUADRILATERAL IS A RECTANGLE, THEN IT HAS CONGRUENT DIAGONALS.
f. A triangular prism has a height of 12 m. What is the surface area of the    
   prism if its bases are right triangles with legs 3 meters and 4 meters?
      (S = Ph + 2B)
g. What is the area of a regular hexagon whose sides are each 12 inches long?
   (round to the nearest square inch) (Clue: Draw the figure)
h. What is the area of a circle with circumference 18 Pi centimeters? Give your
   answer in terms of pi.
i. Find the volume, to the nearest tenth, of a pyramid that is 6 feet
tall and whose base is an equilateral triangle with sides each 10 feet
long.

a.

The total length of two 7-foot sections is 14 feet. The difference
from 11 feet 9 inches is

    (14 + 0/12) - (11 + 9/12) = 2 + 3/12

so the amount of the overlap is 2 feet 3 inches.


b.

The pentagon is composed of five isosceles triangles with base 7 and
height 4.8. Each triangle has area

    7 * 4.8/2 = 7 * 2.4
              = 16.8

and we have

    5 * 16.8 = 84

so the total area is 84 square millimeters.


c.

If solids A and B are similar and the ratio of their heights is 6:15,
then the ratio of their volumes is

    6^3 : 15^3 = 216 : 3375.

To scale down B's volume of 250, we calculate

    216 * 250 / 3375 = 54000 / 3375
                     = 16

to conclude that the volume of solid A is 16 cubic meters.


d.

The sum of the area of the two bases is

    3*4 = 12.

Since the hypotenuse of each base is

    sqrt(3*3 + 4*4) = sqrt(9 + 16)
                    = sqrt(25)
                    = 5

the perimeter of each base is

    3 + 4 + 5 = 12.

We calculate

    Ph + 2B = 12 * 12 + 12
            = 144 + 12
            = 156

to conclude that the surface area of the prism is 156 square meters.


e.

If all squares are rectangles and every rectangle has congruent
diagonals, then, by transitive closure, all squares have congruent
diagonals.

Such a deduction is described by the following syllogism.

    p -> q
    q -> r
    ------
    p -> r


f.

This question is identical to question d above, so the answer is again
156 square meters.


g.

This hexagon is composed of six equilateral triangles, each with side
12 inches. Each such triangle can be bisected into two right triangles
of base 6 and hypotenuse 12. The height of each equilateral triangle
is therefore

    sqrt(12*12 - 6*6) = sqrt(144 -  36)
                      = sqrt(108).

Since each equilateral triangle has area

    6 * sqrt(108)

we calculate

    6 * 6 * sqrt(108) = 36 * 10.392
                      = 374.123

to conclude that the hexagon's area, rounded off to the nearest unit,
is 374 square inches.


h.

If

    2 * pi * radius = 18 pi

then

    radius = 18 / 2
           = 9.

The area is therefore

    pi * radius * radius = 9 * 9 * pi
                         = 81 pi

square centimeters.


i.

The volume of a pyramid is one third its height times the area of its base.

The base of this pyramid can be bisected into two right triangles of
base 5 and hypotenuse 10. The area of the base is therefore

    5 * sqrt(10*10 - 5*5) = 5 * sqrt(100 - 25)
                          = 5 * sqrt(75)
                          = 43.30

Thus, we calculate

    6 / 3 * 43.30 = 2 * 43.30
                  = 86.60

to conclude that the volume of the pyramid, rounded to the nearest
tenth, is 86.6 cubic feet.


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