Of a parabolic arch and is to have a span of 100 feet. The height of the arch a distance of 40 feet from the center is to be 10 feet. Find the height of the arch at its center.

y = -0.11x^2 + 1.111x

y = 28 ft   Height at center

Step-by-step explanation:

Given:-

- The span of the arc is = 100 ft

- The height of the arch is 40 ft at 10 ft from center.

Find:-

- The equation of parabolic arch and the height of the arch at center.

Solution:-

- We will take the height y as a function of width x of the parabolic arch. The general equation of the arch is such that it passes through origin. The equation is given in the form as:

                               y = f(x) = ax^2 + bx

Where,

           a, b, and c are constants to be determined.

- We will use the condition i.e the span of entire arch is 100 ft. So we could say that y = 0 for x = 100 ft. Then we have:

                               0 = f(100) = a(100)^2 + b(100)   1

- Using second condition i.e y = 10 ft at 40 ft from center. Since, due to symmetry we know that center lies at x = 50 ft. Then y = 10 ft at x = 10 ft. The condition can be expressed in the form:

                               10 = f(10) = a(10)^2 + b(10)   2

- Solving the 2 Equations simultaneously, we have:

                               0 = a(100)^2 + b(100)

                               10*10 = a*10*(10)^2 + b(10)*10

                               100 = a(10)^3 + b(100)

- Subtract both equations:

                               100 = a*(10^3 - 100^2)

                               a = 100 / ( 1000 - 10000)

                               a = -0.11

- Then using a = -0.11 evaluate b:

                               -1.11 + 10b = 10

                                b = 11.11 / 10 = 1.111

- The equation of the parabola is:

                                y = -0.11x^2 + 1.111x                                

-The height of the arch at center where x = 50 ft.

                                y = -0.11(50)^2 + 1.111(50)

                                y = -27.5 + 55.5

                                y = 28 ft                                                        

- The height of the parabolic arch at center is given as y = 28 ft.

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step-by-step explanation:

d. $180

step-by-step explanation:

if the dimensions of the room is 9ft by 12 ft, we have the area of the floor as,

so the area of the floor is 108 square feet, converting that into square yards we get,

the cost of carpet is $15 per square yard, therefore the cost of carpet for 12 square yards is,

hence, the cost of carpet is $180.

assuming the dimensions of the room be,

15 ft by 12 ft

so, the area of the floor of that room is,

we can convert 180 square feet into square yards, we know that,

1 sq ft = 0.111 sq yard

so, 180 square feet is,

so the area of floor is 20 square yards.

now, the cost of carpet is 15 dollars per square yard, therefore the cost of carpet for 20 square yards is,

so, it will cost $300.