A large rectangular movie screen in an Imax Theater has an area of 9975 square feet. Find the dimensions of the screen if it is 10 feet longer than it is wide.
width ----- x ft
length = x+10 ft
x(x+10) = 9975
x^2 + 10x - 9975 = 0
intuition:
looks like the two roots are 10 apart
so I took √9975 = 99.87... or appr 100
guessed at 95 and 105, sure enough 95(105) = 9975 , so ....
(x+105)(x-95) = 0
since the width cannot be negative
x = 95
the screen is 95 ft by 105ft
of course we could have solved the equation just using the formula.
To find the dimensions of the movie screen, we can set up a quadratic equation based on the given information.
Let's assume that the width of the movie screen is x feet. Then, according to the problem, the length of the movie screen is 10 feet longer than the width, so the length would be (x + 10) feet.
The area of a rectangle is calculated by multiplying the length by the width. In this case, the area is given as 9975 square feet. So we have the equation:
x * (x + 10) = 9975
Expanding the equation, we get:
x^2 + 10x = 9975
Rearranging the equation to standard quadratic form, we have:
x^2 + 10x - 9975 = 0
Now, we can solve this quadratic equation to find the dimensions of the movie screen. We can either factorize, complete the square, or use the quadratic formula.
Let's use the quadratic formula to solve for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
Comparing the quadratic equation to standard form, we have:
a = 1, b = 10, c = -9975
Substituting the values into the quadratic formula, we get:
x = (-10 ± √(10^2 - 4 * 1 * -9975)) / (2 * 1)
Simplifying further:
x = (-10 ± √(100 + 39900)) / 2
x = (-10 ± √(40000)) / 2
x = (-10 ± 200) / 2
Now we have two possible solutions:
x = (-10 + 200) / 2 = 190 / 2 = 95
or
x = (-10 - 200) / 2 = -210 / 2 = -105
Since the width cannot be negative, we ignore the negative solution.
Therefore, the width of the movie screen is 95 feet, and the length would be (95 + 10) = 105 feet.
So, the dimensions of the screen are 95 feet by 105 feet.