HD Televisions are everywhere, literally. Since the phase out of the old Cathode Ray Tube (CRT) televisions, the way we watch has grown exponentially. Every way from size, 3-D, thickness, location, Wi-Fi, curved, and the number of TV’s. We are living in a time where there is no limit to the number of possibilities.
Many of your teachers use 2 to 3 monitors for work.
Gamers will use multiple screens if they happen to be in the same place.
Restaurants and individuals have TVs mounted everywhere tuned into the current sporting events or news of the day. Yes I mean everywhere, even in the bathrooms.
You have two tasks for this portfolio. Be sure to show all work, save your document, and upload your document to the dropbox in Unit 6 Lesson 10 on page 2.
Task 1 (10 pts): Determine the missing measurements for each TV. You are looking for the width, height, or diagonal. You must show all your work and fill in the table. Remember, TVs are given their size by the diagonal length. (32” means 32 inches)
32” TV height: 16” width: _____
_____ TV height: 34” width: 61”
60” TV height: 30” width: _____
_____ TV height: 20” width: 35”
52” TV height: _____ width: 45”
TV Size (Diagonal)
Height
Width
32”
16”
34”
61”
60”
30”
20”
35”
52”
45”
Task 2 (6 pts): If a TV has the following dimensions 48” wide, 27” height, and 55” diagonal. Respond to the following questions.
If the TV is 5” wider and 3” higher, what is the new diagonal measurement? (Show work)
If the TV is 3” wider and 5” higher, what is the new diagonal measurement? (Show work)
Depending on the previous responses, why are the diagonal measurements the same or different?
To determine the new diagonal measurement if the TV is 5" wider and 3" higher, we can use the Pythagorean theorem. The formula for the Pythagorean theorem is a^2 + b^2 = c^2, where a and b are the lengths of the sides of a right triangle and c is the length of the hypotenuse (in this case, the diagonal of the TV).
Given that the original width is 48", the original height is 27", and the original diagonal is 55", we can set up the equation as follows:
48^2 + 27^2 = c^2
2304 + 729 = c^2
3033 = c^2
c ≈ √3033
c ≈ 55.03
Therefore, the new diagonal measurement when the TV is 5" wider and 3" higher is approximately 55.03".
To determine the new diagonal measurement if the TV is 3" wider and 5" higher, we can use the same process as above.
Given that the original width is 48", the original height is 27", and the original diagonal is 55", we can set up the equation as follows:
(48+3)^2 + (27+5)^2 = c^2
51^2 + 32^2 = c^2
2601 + 1024 = c^2
3625 = c^2
c ≈ √3625
c ≈ 60.21
Therefore, the new diagonal measurement when the TV is 3" wider and 5" higher is approximately 60.21".
The reason the diagonal measurements are different in these two scenarios is because the change in dimensions (width and height) affects the length of the hypotenuse (diagonal length) in a right triangle. The diagonal measurement is directly related to the width and height of the TV, so any change in those dimensions will result in a different diagonal length.
are you sure?
Apologies for the confusion earlier. Let's correct the process and work on the tasks you provided.
**Task 1: Determine the missing measurements for each TV.**
We will use the aspect ratio typically used for most modern TVs, which is 16:9. This means the ratio of width to height is 16 to 9. For any given diagonal, we can find the width and height using the following formulas, which are derived from the Pythagorean theorem where:
The diagonal^2 = width^2 + height^2
And considering the aspect ratio, we relate the height (H) to the width (W) as:
H = (9/16) * W or W = (16/9) * H
Now, let’s solve the unknowns:
1. 32” TV height: 16” width: _____
Using the ratio, we can find the width:
H = (9/16) * W
16 = (9/16) * W
W = 16 * (16/9)
W = 256/9
W ≈ 28.44"
Fill in this table:
TV Size (Diagonal) | Height | Width
---|---|---
32” | 16” | 28.44”
2. _____ TV height: 34” width: 61”
Using the ratio, we can find the diagonal:
H = (9/16) * W
34 = (9/16) * 61
Diagonal^2 = 34^2 + 61^2
Diagonal ≈ √(1156 + 3721)
Diagonal ≈ √4877
Diagonal ≈ 69.84"
Fill in this table:
TV Size (Diagonal) | Height | Width
---|---|---
69.84” | 34” | 61”
3. 60” TV height: 30” width: _____
Using the ratio, we can find the width:
H = (9/16) * W
30 = (9/16) * W
W = 30 * (16/9)
W ≈ 53.33"
Fill in this table:
TV Size (Diagonal) | Height | Width
---|---|---
60” | 30” | 53.33”
4. _____ TV height: 20” width: 35”
Using the ratio, we can find the diagonal:
H = (9/16) * W
20 = (9/16) * 35
Diagonal^2 = 20^2 + 35^2
Diagonal ≈ √(400 + 1225)
Diagonal ≈ √1625
Diagonal ≈ 40.31"
Fill in this table:
TV Size (Diagonal) | Height | Width
---|---|---
40.31” | 20” | 35”
5. 52” TV height: _____ width: 45”
Using the ratio, we can find the height:
W = (16/9) * H
45 = (16/9) * H
H = 45 * (9/16)
H ≈ 25.31"
Fill in this table:
TV Size (Diagonal) | Height | Width
---|---|---
52” | 25.31” | 45”
**Task 2: If a TV has the following dimensions 48” wide, 27” height, and 55” diagonal.**
1. If the TV is 5” wider and 3” higher, what is the new diagonal measurement?
The original TV dimensions already form a valid right triangle, so adding to both the width and height will form a new right triangle with a larger diagonal:
New width: 48" + 5" = 53"
New height: 27" + 3" = 30"
New diagonal^2 = 53^2 + 30^2
New diagonal^2 = 2809 + 900
New diagonal^2 = 3709
New diagonal ≈ √3709
New diagonal ≈ 60.90"
2. If the TV is 3” wider and 5” higher, what is the new diagonal measurement?
New width: 48" + 3" = 51"
New height: 27" + 5" = 32"
New diagonal^2 = 51^2 + 32^2
New diagonal^2 = 2601 + 1024
New diagonal^2 = 3625
New diagonal ≈ √3625
New diagonal ≈ 60.21"
Therefore, the diagonal measurements are different because the changes to width and height are different in each scenario, and these changes result in different diagonal lengths due to the Pythagorean theorem.
To determine the missing measurements for each TV, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Task 1:
For the 32" TV, we are given the height as 16". To find the width, we can use the Pythagorean theorem:
width^2 + 16^2 = 32^2
width^2 + 256 = 1024
width^2 = 1024 - 256
width^2 = 768
width = sqrt(768) ≈ 27.7"
So, the width of the 32" TV is approximately 27.7".
For the second missing measurement, we are given the height as 34" and the width as 61". To find the diagonal, we can use the Pythagorean theorem:
34^2 + 61^2 = diagonal^2
1156 + 3721 = diagonal^2
4877 = diagonal^2
diagonal = sqrt(4877) ≈ 69.9"
So, the diagonal of the second TV is approximately 69.9".
For the 60" TV, we are given the height as 30". To find the width, we can use the Pythagorean theorem:
width^2 + 30^2 = 60^2
width^2 + 900 = 3600
width^2 = 3600 - 900
width^2 = 2700
width = sqrt(2700) ≈ 51.96"
So, the width of the 60" TV is approximately 51.96".
For the fourth missing measurement, we are given the height as 20" and the width as 35". To find the diagonal, we can use the Pythagorean theorem:
20^2 + 35^2 = diagonal^2
400 + 1225 = diagonal^2
1625 = diagonal^2
diagonal = sqrt(1625) ≈ 40.31"
So, the diagonal of the fourth TV is approximately 40.31".
For the 52" TV, we are given the width as 45". To find the height, we can use the Pythagorean theorem:
height^2 + 45^2 = 52^2
height^2 + 2025 = 2704
height^2 = 2704 - 2025
height^2 = 679
height = sqrt(679) ≈ 26.06"
So, the height of the 52" TV is approximately 26.06".
Here is the filled table:
TV Size (Diagonal)
Height
Width
32”
16”
27.7”
34”
61”
60”
30”
51.96”
20”
35”
52”
26.06”
45”
Task 2:
If a TV has the dimensions 48” wide, 27” height, and 55” diagonal, we can use the Pythagorean theorem to find the missing measurements in each question.
If the TV is 5” wider, the new width would be 48 + 5 = 53”. If it is 3” higher, the new height would be 27 + 3 = 30”.
To find the new diagonal measurement, we can use the Pythagorean theorem:
53^2 + 30^2 = diagonal^2
2809 + 900 = diagonal^2
3709 = diagonal^2
diagonal = sqrt(3709) ≈ 60.94"
So, the new diagonal measurement is approximately 60.94".
If the TV is 3” wider, the new width would be 48 + 3 = 51”. If it is 5” higher, the new height would be 27 + 5 = 32”.
To find the new diagonal measurement, we can use the Pythagorean theorem:
51^2 + 32^2 = diagonal^2
2601 + 1024 = diagonal^2
3625 = diagonal^2
diagonal = sqrt(3625) ≈ 60.21"
So, the new diagonal measurement is approximately 60.21".
The diagonal measurements are different in the two scenarios because changing the width and height of the TV affects the proportions of the triangle formed by the width, height, and diagonal. The Pythagorean theorem shows that the diagonal is the hypotenuse of a right triangle formed by the width and height, so changing either the width or the height will result in a different diagonal measurement.