can you help?

Question

1) t^3<9t^2

2)The side of a large tent is in the shape of an isosceles triangle whose area is 54 ft^2 and whose base is 6 feet shorter than twice its height. Find the height and the base of the tent.

3) A decorator plans to place a rug in a room 9 m by 12 m in such a way that a uniform strip of flooring around the rug will remain uncovered. If the rug is to cover half of the floor space, what should the dimensions of the rug be?


Thank you for your help.

Answer 1

1) t^3<9t^2
t < 9
except for t = 0.( t=0 is not a solution )

Answer 2

1)

t^3 < 9t^2

t^3 - 9t^2 <0

t^2(t - 9) < 0

either t <0 or t < 9

t lies in the interval (-∞ , 9]

2)

Let ABC is isosceles triangle, where BC is the base.

Draw perpendicular AD from A to BC.

given that BC = 2AD - 6, where BC = base and AD = h

so b = 2h - 6

area = bh/2

substituting b value in terms of h

54 = (2h-6)h/2

108 = 2h^2 - 6h

2h^2 - 6h - 108 = 0

divide by 2

h^2 - 3h - 54 = 0

(h - 9)(h + 6) = 0

so h = 9 (ignoring negative value )

so b = 2*9 - 6 = 12

so base = 12 ft and height, h = 9 ft

3)

area of the room = 9 * 12 = 108 m^2

so area of the rug = 108/2 = 54 m^2

let the uniform space to be uncovered = x m

so dimensions of the rug = (12 - 2x) and (9 -2x)

so (12 - 2x)(9-2x) = 54

108 - 24x - 18x + 4x^2 = 54

4x^2 - 42x +54 = 0

divide by 2

2x^2 - 21x + 27 = 0

2x^2 - 18x - 3x + 27 = 0

2x(x - 9) - 3(x - 9) = 0

(x-9)(2x-3) = 0

so x = 9 or 3/2

9 is invalid value, since width of space can not excced the width of the room.

so x = 3/2 = 1.5

rug dimensions are (12 - 3) and (9-3) => 9 m by 6 m

Answer 3

1) t^3 > 9t^2.
since t^2 is never negative and if t=0, the inequality is false, we can divide both sides by t^2 and maintain the inequality.
t > 9

2) height x, base 2x-6. A = 1/2x(2x-6) = x(x-3) = x^2-3x = 54
x^2-3x-54 = 0
(x-9)(x+6) = 0
x=9. base = 12.

3) let x be the width of the strip of flooring.
so each dimension is reduced by 2x (floor on all sides).
(9-2x)(12-2x) = 9*12/2
4x^2 - 42x + 108 = 54
4x^2-42x+54 = 0
2x^2-21x+27 = 9
(2x-3)(x-9) = 0
x=9 means negative rug dimensions
x=3/2
rug is 6 by 9