The diagonal of a TV set is 26 inches long. Its length is 14 inches more than the height. Find the dimensions of the TV set.
height ---- x
length ---- x+14
x^2 + (x+14)^2 = 26^2
2x^2 + 28x + 196 = 676
x^2 + 14x = 240
completing the square ...
x^2 + 14x + 49 = 240+49
(x+7)^2 = 289
x+7 = √289 = 17
x = 10
the TV is 10 inches by 24 inches
(odd size)
check:
10^2 + 24^2
= 100 + 576
= 676
= 26^2
My answer is correct
13
Why did the TV set go to the doctor?
Because it was feeling a little diagonal!
But hey, let me calculate those dimensions for you.
Let's say the height of the TV set is "h" inches. According to the problem, the length is 14 inches longer, so it would be "h + 14" inches.
Now, we can use the Pythagorean theorem to find the diagonal. The formula is D^2 = h^2 + L^2, where D is the diagonal.
Plugging in the given values:
26^2 = h^2 + (h + 14)^2
676 = h^2 + h^2 + 28h + 196
2h^2 + 28h - 480 = 0
h^2 + 14h - 240 = 0
Solving this quadratic equation, we get two possible solutions: h = 10 or h = -24.
Since the height can't be negative, the only valid solution is h = 10 inches.
Therefore, the dimensions of the TV set are height = 10 inches and length = 10 + 14 = 24 inches.
Let's assume the height of the TV set is x inches.
According to the given information, the length of the TV set is 14 inches more than the height, so it can be written as x + 14 inches.
Using the Pythagorean theorem, we know that the square of the diagonal is equal to the sum of the squares of the other two sides. In this case, it is:
diagonal^2 = height^2 + length^2
Substituting the given values, we have:
26^2 = x^2 + (x + 14)^2
Expanding the equation:
676 = x^2 + x^2 + 28x + 196
Combining like terms:
2x^2 + 28x + 196 = 676
Simplifying the equation:
2x^2 + 28x - 480 = 0
Dividing both sides of the equation by 2:
x^2 + 14x - 240 = 0
Factoring the quadratic equation:
(x + 24)(x - 10) = 0
Setting each factor equal to zero:
x + 24 = 0 or x - 10 = 0
Solving for x:
x = -24 or x = 10
Since height cannot be negative, the height of the TV set is 10 inches.
Substituting x = 10 back into the equation for the length:
Length = x + 14 = 10 + 14 = 24 inches
Therefore, the dimensions of the TV set are height = 10 inches and length = 24 inches.
To find the dimensions of the TV set, we can set up a system of equations based on the given information.
Let's assume the height of the TV set is represented by h inches.
According to the given information, the length of the TV set is 14 inches more than the height. Therefore, the length would be h + 14 inches.
We also know that the diagonal of the TV set is 26 inches.
Using the Pythagorean theorem, we can relate the height, length, and diagonal of the TV set:
(diagonal)^2 = (height)^2 + (length)^2
Substituting the given values, we have:
26^2 = h^2 + (h + 14)^2
676 = h^2 + (h + 14)^2
Expanding the equation, we get:
676 = h^2 + (h^2 + 28h + 196)
Combining like terms:
2h^2 + 28h + 196 = 676
2h^2 + 28h + 196 - 676 = 0
2h^2 + 28h - 480 = 0
Dividing the equation by 2 to simplify:
h^2 + 14h - 240 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula.
Factoring the equation, we have:
(h - 8)(h + 30) = 0
From this, we get two possible solutions for h:
h - 8 = 0 OR h + 30 = 0
h = 8 OR h = -30
Since the dimensions of a TV cannot be negative, we discard the second solution and solve for h = 8.
Therefore, the height of the TV set is 8 inches.
Now, let's find the length of the TV set using the equation:
Length = Height + 14
Length = 8 + 14
Length = 22 inches
So, the dimensions of the TV set are 8 inches (height) and 22 inches (length).