High-definition (HD) televisions today have a 16:9 aspect ratio (width to height). The advertised screen size is equal to the screen diagonal.
Allowing for 2 inches of Styrofoam padding on all sides of the TV, what is the smallest possible length and width of a shipping box for a 75-inch HD TV?
HELPP
64 by 36
75 by 44
68 by 40
60 by 45
To determine the smallest possible length and width of a shipping box for a 75-inch HD TV, we will need to account for the diagonal measurement along with the 2 inches of Styrofoam padding on all sides.
First, let's calculate the height of the TV using the 16:9 aspect ratio. The height can be found by dividing the screen width by the aspect ratio:
Height = Screen Width / (16/9)
Height = Screen Width * (9/16)
For a 75-inch TV, the width is 75 inches. Substituting this value in the formula, we have:
Height = 75 * (9/16)
Height ≈ 42.1875 inches
Next, let's calculate the diagonal length of the TV including the padding. This can be calculated using the Pythagorean theorem:
Diagonal Length = √(Height^2 + Width^2)
Diagonal Length = √((42.1875+4)^2 + (75+4)^2)
Diagonal Length = √(46.1875^2 + 79^2)
Diagonal Length = √(2129.513 + 6241)
Diagonal Length ≈ √8370.513
Diagonal Length ≈ 91.448 inches
Finally, we add the 2 inches of padding on all sides to the diagonal length:
Diagonal Length with Padding = Diagonal Length + 2 + 2
Diagonal Length with Padding = 91.448 + 2 + 2
Diagonal Length with Padding ≈ 95.448 inches
Therefore, the smallest possible length and width of a shipping box for a 75-inch HD TV with 2 inches of Styrofoam padding on all sides would be approximately 95.448 inches by 42.1875 inches.
None of the given options match this calculation, so none of the provided options are correct.
To determine the minimum length and width of the shipping box for the 75-inch HD TV, we need to consider the diagonal length of the TV and add the additional padding on all sides.
First, let's calculate the diagonal length of the TV using the aspect ratio of 16:9. We can do this using the Pythagorean theorem, where the diagonal length squared is equal to the sum of the squares of the width and height.
Given that the width to height ratio is 16:9, we can assume the width is 16x and the height is 9x (where x is a common factor).
Using the Pythagorean theorem:
Diagonal length squared = (width squared) + (height squared)
75 squared = (16x) squared + (9x) squared
5625 = 256x^2 + 81x^2
5625 = 337x^2
x^2 = 5625/337
x^2 ≈ 16.7
Taking the square root of both sides, we find:
x ≈ √(16.7)
x ≈ 4.08
Now, we can find the actual width and height of the TV by multiplying x by 16 and 9, respectively:
Width ≈ 4.08 * 16 ≈ 65.28 inches
Height ≈ 4.08 * 9 ≈ 36.72 inches
Next, we need to add 2 inches of Styrofoam padding to each side. Therefore, the minimum length and width of the shipping box would be:
Length = Width + 2 * 2 = 65.28 + 2 * 2 = 65.28 + 4 = 69.28 inches
Width = Height + 2 * 2 = 36.72 + 2 * 2 = 36.72 + 4 = 40.72 inches
Therefore, the smallest possible length and width of the shipping box for a 75-inch HD TV with 2 inches of Styrofoam padding on all sides is approximately 69.28 inches by 40.72 inches.
So the correct answer is 68 by 40, rounding to the nearest whole number.
With screen width 16x and height 9x, you have
(16x)^2 + (9x)^2 = 75^2
So add 4 in to both width and height