Respuesta :
Answer:
x ∈ (3 - sqrt(3) , 3 + sqrt(3) )
Distance from both ends from ground is 1.1 ft
Step-by-step explanation:
Given:
- The parabolic shape of the entrance of the room is given by:
f(x) = - x^2 + 6x + 1
Find:
- How far from the sides of the arch is its height at least 7 feet?
Solution:
- We are asked to find for what values of x is the height f(x) is greater than 7.
So we construct and inequality:
f(x) = - x^2 + 6x + 1 > 7
- Solve the inequality:
x^2 - 6x + 6 < 0
- < 0 , signs indicates the domain of x for which the function represented by and inequality is below x-axis or is negative:
- Complete squares we get:
(x - 3 )^2 - 3 = 0
(x - 3 )^2 = 3
(x-3) = +/- sqrt(3)
x_1 = 3 - sqrt(3)
x_2 = 3 + sqrt(3)
- So the domain of values for which the height of function is 7 feets is:
x ∈ (3 - sqrt(3) , 3 + sqrt(3) )
- Now compute the intercept of arc with the ground:
f(x) = - x^2 + 6x + 1 = 0
x^2 - 6x - 1 = 0
x = 0.17 , 5.83
- distances from the sides:
Left side: 3 - sqrt(3) - 0.17 = 1.1 ft
Right side: 5.83 - 3 - sqrt(3) = 1.1 ft
Hence the distance from the sides from ground is 1.1 ft
