Week 9

course Mth 152

??????????????assignment #022

Your work has been received. Please scroll through the document to see any inserted notes (inserted at the appropriate place in the document, in boldface) and a note at the end. The note at the end of the file will confirm that the file has been reviewed; be sure to read that note. If there is no note at the end, notify the instructor through the Submit Work form, and include the date of the posting to your access page.

022. `query 22

Liberal Arts Mathematics II

07-29-2007

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15:57:05

**** query 9.3.12 area of parallelogram **** What is the area of the given parallelogram and how did you obtain it?

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RESPONSE -->

A = B * H

A= 4 * 2.5

A = 10

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16:00:04

The formula for the area of a parallelogram is

A = bh ( area = base times height)

From the figure the base is 4 and the altitude is 2.5 so we have

A = 4 * 2.5 = 10.

The area is 10 in.

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RESPONSE -->

Ok.

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16:01:51

**** query 9.3.18 area of trapezoid **** What is the area of the given trapezoid and how did you obtain it?

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RESPONSE -->

A = .5 * H (b + B)

A = .5 * 3 (4 + 5)

A = .5 * 3 (9)

A = 13.5

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16:07:41

The formula for finding the area of a trapezoid is A = 1/2h ( b + B )

h = 3, b = 4, B = 5

A = 1/2 (3) (4 + 5)

A = 1/2 (3) (9)

A = 1/2 (27)

A = 13.5

The area is 13.5

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RESPONSE -->

Ok.

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16:14:19

**** query 9.3.24 dim of rect with lgth 20 more than wdth and perimeter 176 **** What are the dimensions of the rectangle?

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RESPONSE -->

P = 2L + 2W

176 = 2(20+W) + 2W

176 = 40 + 4W

136 = 4W

34 = W

W = 34 L = 54

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16:14:30

** The perimeter of a rectangle is P = 2l + 2w, where l and w are length and width.

From the given information

Length = 20 + w

Perimeter = 176

This gives us the equation

176 = 2 (20 + w) + 2w

which we proceed to solve for w:

176 = 40 + 2w + 2w by the Distributive Law. Simplifying we get

176 = 40 + 4w

Subtract 40 from both sides to get

136 = 4w

Divide both sides by 4 to get

w = 34

l = w + 20 so

l = 34 + 20 = 54

We have

Length = 54 and

Width = 34

Checking, we have perimeter = 2 * length + 2 * width so we should have

176 = 2(54) + 2(34).

The right-hand side does give us 176 so the solution checks out. **

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RESPONSE -->

Ok.

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16:17:56

**** query 9.3.48 trap bases x, x+4 alt 3 area 30 **** What is the value of x?

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RESPONSE -->

A = .5 * H (b + B)

30 = .5 * 3 (X + X+4)

30 = 1.5 (X + X+4)

30 = 1.5X + 1.5X + 6

30 = 3X + 6

24 = 3X

8 = X

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16:18:18

The formula for finding the area of a trapezoid is A = 1/2h ( b + B). We have

A = 30

h = 3

B = x + 4

b = x

30 = 1/2(3) ( x + x+4)

30 = 1.5 ( x+x+4)

30 = 1.5x + 1.5x + 6

30 = 3x + 6 Subtract 6 from both sides

24 = 3x Divide both sides by 3

x = 8

The answer is 8.

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RESPONSE -->

Ok.

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16:32:24

**** query 9.3.54 $60 to paint ceiling of 9 x 15 rm, how much to paint if dimensions 18 x 30 **** What is the cost for the second ceiling? **** How did you use the results of exercise 53 to obtain your result?

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RESPONSE -->

Sides double so area goes up by 4;

60 * 4 = $240

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16:32:33

The sides doubled from 9 ft. to 18 ft. and from 15 ft. to 30 ft. When the sides are doubled the area increases by a factor of 4. So the cost is

$60 * 4 = $240

The cost for the second ceiling is $240

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RESPONSE -->

Ok.

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16:43:24

**** query 9.3.60 triangle alt 9 on top of 10 x 4 rect; parallelogram alt 3 under **** What is the area of the given figure and how did you obtain your result?

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RESPONSE -->

.5 * B * H

A = .5 * (10) * (9)

A = .5 * (90)

A = 45

A = LW

A = (10) * (4)

A = 40

A = BH

A = (10) * (3)

A = 30

45 + 40 + 30 = 115

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16:43:39

** Here's a solution by a student from a previous semester:

I found the area of each figure and then added those three results together.

Area of Triangle = 1/2bh

A = 1/2 (10) (9)

A = 1/2 (90)

A = 45

Area of Rectangle = lw

A = (10) (4)

A = 40

Area of Parallelogram = bh

A = (10) (3)

A = 30

45 + 40 + 30 = 115

Area = 115 **

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RESPONSE -->

Ok.

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16:46:09

**** query 9.3.67 26 m diam circle inscribed in sq; area outside circle **** What is the area of the shaded region and how did you obtain it?

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RESPONSE -->

A = 3.14 R^2

A = 3.14 * 13^2

A = 530.7 m^2

A = (26)^2 = 676

676m^2 - 531m^2 = 145m^2

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16:46:19

** The circle has diameter 26 m so its radius is 13 m and its area is

A = pi r^2 = pi * (13 m)^2 = 169 pi m^2 = 531 m^2.

The area of the square is the square of its side

A = (26 m)^2 = 676 m^2.

The area of the shaded region is the difference

}

676 m^2 - 531 m^2 = 145 m^2. **

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RESPONSE -->

Ok.

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16:52:25

**** query 9.3.72 10, 12, 14 in pizzas for 11.99, 13.99, 14.99 **** Which pizza is the best buy and how did you

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RESPONSE -->

A = 3.14 * (5)^2 = 78.5

11.99 / 78.5 = 15.3

A = 3.14 * (6)^2 = 113.0

13.99 / 113.0 = 12.4

A = 3.14 * (7)^2 = 153.9

14.99 / 153.9 = 9.7....Best

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16:52:44

Student Solution: The thickness is about the same for all three pizzas so the amount of pizza can be measured by its area. I therefore found the area of each.

A = 3.14 * r^2 and radius is 1/2 the circumference. So we get areas

A = 3.14 (5)^2 = 78.5

A = 3.14 (6)^2 = 113.04

A = 3.14 (7)^2 = 153.86

I then divide the prices and these answers to get the price per square inch.

$11.99 / 78.5 = 15.3

$13.99 / 113.04 = 12.4

$14.99 / 153.86 = 9.7

Since 9.7 is the least, and since this is the result for the 14 inch pizza, the 14 in. pizza for $14.99 is the best buy.

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RESPONSE -->

Ok.

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16:52:49

**** Query Add comments on any surprises or insights you experienced as a result of this assignment.

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RESPONSE -->

None

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????B???K??????assignment #023

023. `query 23

Liberal Arts Mathematics II

07-29-2007

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20:40:35

**** query 9.4.6 ABC, DEF transversed by EOB at rt angles; OB = EO; show triangles ABO and DOF congruent.

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RESPONSE -->

AOB and FOE are equal since they are vertical angles, so since 2 sides of both triangles are equal you can use the side angle side property to show congruency.

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20:40:58

SAS: Angle AOB and Angle FOE are equal because they are vertical angles, so we have 2 sides and the included angle of triangle AOB equal, respectively, to 2 sides and the included angle of triangle FOE. Thus, the Side-Angle-Side property holds that triangle AOB is congruent to triangle FOE.

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RESPONSE -->

Ok.

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20:43:27

**** Explain the argument you used to show that the triangles were congruent.

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RESPONSE -->

Side angle side property

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20:48:18

**** query 9.4.18 ACB and QPR similar triangles, C and P rt angles, A=42 deg **** List the measures of the three angles of each triangle and explain how you obtained each.

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RESPONSE -->

A = 42, C = 90: 180-(A+C) = 48 = B

P = 90, 90/90 = 42/42 = Q, 4320/90 = 48 = R

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20:49:15

It is given that Angle A = 42 deg. and Angle C = 90 deg. Since all three angles must add up to equal 180 then Angle B = 48 deg.

In the second triangle, Angle P must equal 90 deg. since it is a right angle.

To find Angle R,

90(48) = 90R sp

4320 = 90R and

48 = R Angle R = 48 deg.

To find Angle Q,

90/90 = Q/42

Q = 42

Angle Q = 42 deg.

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RESPONSE -->

Ok.

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20:53:07

**** query 9.4.24 similar triangles, corresp sides a, b, 75; 10, 20, 25 **** What are the lengths of sides a and b and how did you obtain each?

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RESPONSE -->

750/25 = A = 30

1500/25 = B = 60

C = 75

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20:53:22

To find a,

75 (10) = 25a

750 = 25a

a= 30

To find b,

75/25 = b/20

1500/25 = 25b/25 so

b = 60.

a = 30, b = 60 and c = 75.

These values are triple the values of the similar triangle.

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RESPONSE -->

Ok.

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20:55:17

**** query 9.4.42 rt triangle a = 7, c = 25, find b **** What is the length of side b and how did you obtain it?

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RESPONSE -->

A^2 + B^2 = C^2

49 + B^2 = 625

B^2 = 576

B = 24

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20:55:31

By the Pythagorean Theorem a^2 + b^2 = c^2. So we have

49 + b^2 = 625 Subtract 49 from both sides to get

b^2 = 576. Take the square root of both sides to get

b = 24.

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RESPONSE -->

Ok.

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20:57:13

**** What does the Pythagorean Theorem say about the triangle as given and how did you use this Theorem to find the length of b?

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RESPONSE -->

The sum of the lengths of the legs squared are equal to the square of the hypotenuse.

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20:57:49

Student Response: It says the sum of the squares of the lengths of the legs is equal to the square of the hypotenuse. I showed that this is true in the previous problem. I squared the legs and they equaled the hyppotenuse squared.

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RESPONSE -->

Ok.

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21:01:33

**** query 9.4.60 m, (m^2 +- 1) / 2 gives Pythagorean Triple **** What Pythagorean Triple is given by m = 5?

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RESPONSE -->

5

(m^2 +1) /2 = (5^2 +1) /2 = 26 /2 = 13

(m^2 -1) /2 = (5^2 -1) /2 = 24 /2 = 12

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21:02:00

** If m = 5 then

(m^2 + 1) / 2 = (5^2 + 1 ) / 2 = 26 / 2 = 13

(m^2 - 1) / 2 = (5^2 - 1 ) / 2 = 24 / 2 = 12

So the Pythagorean triple is 5, 12, 13.

We can verify this:

5^2 + 12^2 should equal 13^2.

5^2 + 12^2 = 25 + 144 = 169.

13^2 = 169.

The two expressions are equal so this is indeed a Pythagorean triple. **

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RESPONSE -->

Ok.

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21:03:00

**** How did you verify that your result is indeed a Pythagorean Triple?

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RESPONSE -->

You could use the numbers inserted into the pythagorean theorem to check.

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21:03:33

Student Answer: The numbers checked out when substituted into the Pythagorean Theorem.

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RESPONSE -->

Ok.

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21:04:58

**** query 9.4.75 10 ft bamboo broken, upper end touches ground 3 ft from stem. **** How high is the break, and how

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RESPONSE -->

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21:05:01

did you obtain your result?

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RESPONSE -->

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21:05:33

** If the break is at height x then the hypotenuse, consisting of the broken part, is at height 10 - x.

The triangle formed by the vertical side, the break and the ground therefore has legs x and 3 and hypotenuse 10-x.

So we have

x^2 + 3^2 = (10-x)^2. Squaring the 3 and the right-hand side:

x^2 + 9 = 100 - 20 x + x^2. Subtracting x^2 from both sides

9 = 100 - 20 x so that

-20 x = -91 and

x = 4.55.

The break occurs at height 4.55 ft and the broken part has length 10 - 4.55 = 5.45, or 5.45 feet. **

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RESPONSE -->

I understand.

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21:05:57

**** How did the Pythagorean Theorem allow you to solve this problem?

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RESPONSE -->

Insert the numbers into the theorem.

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21:06:03

I substituted the numbers into the Pythagorean Theorem.

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RESPONSE -->

Ok.

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21:06:47

**** query 9.4.84 isosceles triangle perimeter 128 alt 48 **** What is the area of the triangle and how did you find

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RESPONSE -->

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21:06:50

it?

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RESPONSE -->

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21:08:26

** This problem is algebraically demanding. Your text might have a slicker way to do this, but the following works:

If the equal sides are x then the base is 128 - 2 x.

The altitude forms a right triangle with half the base and one of the equal sides. The sides of this right triangle are therefore 48, 1/2 (128 - 2x) = 64 - x, and x.

The right angle is formed between base and altitude so x is the hypotenuse.

We therefore have

48^2 + (64 - x)^2 = x^2 so that

48^2 + (64 - x) ( 64 - x) = x^2 or

48^2 + 64 ( 64-x) - x(64 - x) = x^2 or

48^2 + 64^2 - 64 x - 64 x + x^2 = x^2 or

48^2 + 64^2 - 128 x + x^2 = x^2. Subtracting x^2 from both sides we get

48^2 + 64^2 - 128 x = 0. Adding 128 x to both sides we get

48^2 + 64^2 = 128 x. Multiplying both sides by 1/128 get have

(48^2 + 64^2) / 128 = x. Evaluating this expression we end up with x = 50.

The base of the triangle is therefore 128 - 2x = 128 - 2 * 50 = 128 - 100 = 28.

So its area is 1/2 b h = 1/2 * 28 * 48 = 672. **

DRV

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RESPONSE -->

I think I got it.

You didn't show the details of your solution on that last one, so I can't tell for sure.

However you're doing OK overall.

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21:11:11

**** Query Add comments on any surprises or insights you experienced as a result of this assignment.

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RESPONSE -->

Geometry isn't really my forte.

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21:11:45

I had some trouble with the last 2 problems.

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RESPONSE -->

Ok.

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o????????f???????

assignment #024

024. `query 24

Liberal Arts Mathematics II

07-29-2007

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21:19:32

**** query 9.5.12 vol of sphere diam 14.8 **** What is the volume of the sphere and how did you obtain it?

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RESPONSE -->

V = 4/3 * 3.14 * (7.4)^3

V = 4/3 * 3.14 * 405.2

V = 1696.4

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21:19:56

I use the formula for finding the volume of a sphere which is 4/3(3.14)(r^3).

Since the diameter is 14.8, the radius is half that which is 7.4.

V = 4/3 * 3.14 * 7.4^3

V = 4/3 * 3.14 * 405.224

V = 1696.54

The volume is 1696.54

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RESPONSE -->

Ok.

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21:22:09

**** query 9.5.18 pyramid 12 x 4 altitude 10 **** What is the volume of the pyramid and how did you find it?

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RESPONSE -->

V = 1/3 * B * H

V = 1/3 * 48 * 10

V = 160

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21:22:51

I used the formula : V = 1/3Bh

The base = 12 * 4 = 48

V = 1/3 * 48 * 10

V = 1/3 * 480

V = 160

The volume is 160ft.^3

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RESPONSE -->

Ok.

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21:25:18

**** query 9.5.24 bottle 3 cm alt 4.3 cm **** What is the volume of the bottle and how did you find it?

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RESPONSE -->

V = 3.14 * (1.5)^2 * 4.3

V = 3.14 * 2.25 * 4.3

V = 30.4

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21:25:59

** The figure is a right circular cylinder with V = 3.14 * r^2 * h

Since the diameter is 3, then the radius is 1.5

V = 3.14 * 1.5^2 * 4.3

V = 3.14 * 2.25 * 4.3

V = 30.38 cm^3 **

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RESPONSE -->

Ok.

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21:32:35

**** query 9.5.36 sphere area 144 `pi^2 **** What are the radius, diameter and volume of the sphere and how did you find them?

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RESPONSE -->

4 Pi R^2 = 144 Pi

R^2 = 36 m^2

R = 6 m

D = 6 * 2 = 12

V = 4/3 Pi * (6)^2 = 48 Pi

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21:33:05

** Sphere area is 4 pi r^2, so we have

4 pi r^2 = 144 pi m^2. Dividing by 4 pi we get

r^2 = 36 m^2. Taking the square root of both sides we get

r = 6 m.

From this we find that the diameter is 2 * 6 m = 12 m and the volume is 4/3 pi * (6 m)^2 = 48 pi m^2. **

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RESPONSE -->

Ok.

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21:39:21

**** query 9.5.48 cone alt 15 rad x vol 245 `pi **** What is the value of x and how did you find your result?

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RESPONSE -->

V = 1/3 Pi R^2 H

3V / (Pi * H) = R^2

R = Sqrt(3V/(pi*H))

R = Sqrt((3*245)/(3.14*15))

R = Sqrt(735/47.1)

R = Sqrt (15.6)

R = 3.9

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21:39:36

** We have V = 1/3 pi r^2 h. To solve for r we multiply both sides by 3 / (pi * h) to get

3 V / (pi * h) = r^2 then take the square root to get

r = sqrt(3 V / ( pi * h). Substituting we get

r = sqrt( 3 * 245 / (3.14 * 15) ) = 3.9 approx. **

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RESPONSE -->

Ok.

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21:40:05

**** query 9.5.51 plane intersects sphere passing 7 in from center forming circle with area 576 `pi **** What is the volume of the sphere and how did you obtain it?

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RESPONSE -->

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21:42:29

** The circle does have radius sqrt(576 in^2) = 24 in. However that is not the radius of the sphere since the plane containing the circle passes 7 in from the center of the sphere. So the center of the circle is not the center of the sphere.

The center of the circle is 7 in from the center of the sphere. Note also that a line from the center of the sphere to the center of the circle will be perpendicular to the plane of the circle.

Thus if you start at the center of the sphere and move the 7 in straight to the center of the circle, then move the 24 in to the rim of the circle, then back to the center of the sphere you will have traced out a right triangle with legs 7 in and 24 in. The hypotenuse of the triangle is the radius R of the sphere.

So we have

R^2 = 7^2 + 24^2 = 625 and

R = 25.

The radius of the sphere is 25 in. **

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RESPONSE -->

I see now.

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21:43:46

**** Query Add comments on any surprises or insights you experienced as a result of this assignment.

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RESPONSE -->

None.

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21:44:19

I don't think that my last answer is correct. Also, in the first problem, the answer should be 1696.54 cm^3, instead of just 1696.54

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RESPONSE -->

Ok.

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&#

Good responses. Let me know if you have questions. &#