How the hell will I solve this Math Q? [ In each of the following cases, x and y are related by a law of the *

Question

* form y is directly proportional to x^n. Find the value of n

a. y is the area of a square and x is the length of one side

b. y is the area of a circle and x is its radius

c. y is the volume of a sphere and x is its radius

d. y is the length of a rectangle of constant area and x is its breadth.

e. y is the radius of a circle and x is its area.

f. y is the length of a line in centimetres and x is its length in millimetres.

g. y is the speed and x is the time taken to travel round one lap of a racing track.

Help me to solve this..Please]

Answer 1

Given: y is directly proportional to x^n. Find the value of n

(a)
Y = Area.
√Y = X
y = X²
→ n = 2

(b)
Y = π r²
Y = πX²
→ n = 2

(c)
Y = 4/3π r^3
Y = 4/3π X^3
→ n = 3

(d)
Y = L * B
Y = L * X
→ n = 1

(e)
X = π r²
X = π Y²
X/π = Y²
(1/π)X = Y²
√[ (1/π)X ] = Y
[ (1/π^½)(X^½) ] = Y
(π^-½)(X^½) = Y
→ n = ½

(f)
Y = L (cm)
X = 10L (mm)
X/10 = L (mm)
Y = X/10
Y = (1/10)X
→ n = 1

(g)
Speed (S) = Distance (D) / Time (T)
Y = S
Y = D/T
Y = D/X
Y = D(X^-1)
→ n = -1

Answer 2

When it says that y is directly proportional to x^n it means that y equals a constant multiplied by x^n. All your questions are just common area/length/volume formulae except that they use, for example, the letter x instead of the letter r for radius etc.

So for part a. the area of a square is just A = S^2 where A is the area and S is the length of the side. Then, replacing A with y and S with x, you get y = x^2. Note that the constant multiplied by x^n is just 1.

Answer 3

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Answer 4

a. y = x^2
b. y = pi*x^2
c. y = 4/3 pi x^3
d. y = A x^(-1)
e. y = (x/pi)^0.5
f. y = x/10
g. y = Cx^(-1)