A geometrical figure Cone with different attributes like base, slant side and height but this problem helps you to calculate height of cone with the help of volume of the cone.

Topic : Height of a Cone

Formula for finding a Volume of cone = 1/3 * ￗ * r² * h

Question : Suppose sand is being poured onto a cone- shaped pile, beginning at time t = 0, at the rate of 29.4 cubic inches per minute. At t = 2 minutes the resulting cone has a diameter of 7″. As the sand continues to be poured, the cone is always similar to its original shape, but it grows in size. Find the height of the cone at t=2minutes(remember to include the base of the pile). Round your answers to the nearest hundredth

Solution :

Rate of increase of volume is 29.9 cubic inches

Hence in the first minute the volume becomes 29.4 cubic inches

So when t = 2 (second minute) Volume = 2 * 29.4 = 58.8 cubic inches

at t = 2 ; V = 1/3 * ￗ * r² * h ; r = 7/2 = 3.5″

So 58.8 = 1/3 * 3.14 * 3.5 * 3.5 * h

58.8 = 12.822 h

h = 58.8/12.822

h = 4.5858

h = 4.59″

Thus after two minutes the pile has a height of 4.59″

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Topic : Geometric Progression

Problem : The fourth term of a G.P. is 9 and the ninth term is 2,187. Find the first 4 terms of the G.P.

Solution :

Let the G.P. be a, ar, ar² , ar³, …

T₄ = ar³ = 9 ———- (1)

T₈ = ar⁸ = 2,187 —— (2)

(2) ¸(1) :   ar⁸ / ar³ = 2187/9

r⁵ = 243

r⁵ = 3⁵

r = 3

Substitute r = 3 into (1):  a(3³) = 9

a = 1/3

Therefore the first four terms of the G.P. are 1/3, 1/3(3), 1/3(3)², 1/3(3)³,

ie 1/3, 1, 3, 9

Topic : Solve for x

Problem : Solve (x+2)/(x-3) = 42/(x+7)

Solution :

Cross multiply,

(x + 2) (x + 7) = 42 (x – 3)

x + 2x + 7x + 14 = 42x -126

x + 9x + 14 = 42x – 126

   – 9x  – 14 = -9x – 14(adding -9x and -14 on both sides)

x = 33x – 140

x – 33x + 140 = 0

x – 28x - 5x + 140 = 0

x(x – 28) - 5(x + 28) = 0

(x-28)(x-5) = 0

x-28 = 0 or x-5=0

x = 28 or x = 5