Yes, Linda would like to install a new screen on the shelf that is 17 inches tall and 50 inches wide. She would like the screen to be the largest size that will fit on the shelf. She knows the diagonal and width of the 4 screens listed below. A: diagonal 15 width 13, B: diagonal 22 width 19, C: diagonal 32 width 28, D: diagonal 40 width 35. Use the Pythagorean theorem to calculate the height of each screen and complete the table.
To calculate the height of each screen using the Pythagorean theorem, we can use the formula:
h^2 = d^2 - w^2
Where h is the height, d is the diagonal and w is the width.
For screen A:
h^2 = (15^2) - (13^2)
h^2 = 225 - 169
h^2 = 56
h = √56
h ≈ 7.48
For screen B:
h^2 = (22^2) - (19^2)
h^2 = 484 - 361
h^2 = 123
h = √123
h ≈ 11.09
For screen C:
h^2 = (32^2) - (28^2)
h^2 = 1024 - 784
h^2 = 240
h = √240
h ≈ 15.49
For screen D:
h^2 = (40^2) - (35^2)
h^2 = 1600 - 1225
h^2 = 375
h = √375
h ≈ 19.36
So, the calculated heights for the screens are approximately:
A: 7.48 inches
B: 11.09 inches
C: 15.49 inches
D: 19.36 inches
To calculate the height of each screen, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Let's calculate the height of each screen using the Pythagorean theorem:
For screen A with a diagonal of 15 inches and a width of 13 inches:
Using the Pythagorean theorem, we have:
Height^2 + Width^2 = Diagonal^2
Height^2 + 13^2 = 15^2
Height^2 + 169 = 225
Height^2 = 225 - 169
Height^2 = 56
Height ≈ √56
Height ≈ 7.48 inches
For screen B with a diagonal of 22 inches and a width of 19 inches:
Using the Pythagorean theorem, we have:
Height^2 + Width^2 = Diagonal^2
Height^2 + 19^2 = 22^2
Height^2 + 361 = 484
Height^2 = 484 - 361
Height^2 = 123
Height ≈ √123
Height ≈ 11.09 inches
For screen C with a diagonal of 32 inches and a width of 28 inches:
Using the Pythagorean theorem, we have:
Height^2 + Width^2 = Diagonal^2
Height^2 + 28^2 = 32^2
Height^2 + 784 = 1024
Height^2 = 1024 - 784
Height^2 = 240
Height ≈ √240
Height ≈ 15.49 inches
For screen D with a diagonal of 40 inches and a width of 35 inches:
Using the Pythagorean theorem, we have:
Height^2 + Width^2 = Diagonal^2
Height^2 + 35^2 = 40^2
Height^2 + 1225 = 1600
Height^2 = 1600 - 1225
Height^2 = 375
Height ≈ √375
Height ≈ 19.36 inches
Based on the calculations, here is the completed table:
Screen | Diagonal | Width | Height
----------------------------------
A | 15 | 13 | 7.48 inches
B | 22 | 19 | 11.09 inches
C | 32 | 28 | 15.49 inches
D | 40 | 35 | 19.36 inches
To determine the largest screen that will fit on the shelf, we can compare the heights of the screens and choose the one with the maximum height. In this case, Screen D has the largest height (19.36 inches), which will fit on a shelf that is 17 inches tall.
To calculate the height of each screen using the Pythagorean theorem, we can use the formula:
height = sqrt(diagonal^2 - width^2)
Let's calculate the height for each screen:
For screen A:
height = sqrt(15^2 - 13^2)
= sqrt(225 - 169)
= sqrt(56)
≈ 7.48 inches
For screen B:
height = sqrt(22^2 - 19^2)
= sqrt(484 - 361)
= sqrt(123)
≈ 11.09 inches
For screen C:
height = sqrt(32^2 - 28^2)
= sqrt(1024 - 784)
= sqrt(240)
≈ 15.49 inches
For screen D:
height = sqrt(40^2 - 35^2)
= sqrt(1600 - 1225)
= sqrt(375)
≈ 19.36 inches
Now we can complete the table:
| Screen | Diagonal | Width | Height |
|--------|----------|-------|--------|
| A | 15 | 13 | 7.48 |
| B | 22 | 19 | 11.09 |
| C | 32 | 28 | 15.49 |
| D | 40 | 35 | 19.36 |
From the table, we can see that the largest screen that will fit on the 17-inch tall shelf is screen C, with a height of 15.49 inches.