It is given that y
is directly proportional to x3.
Find the percentage increase in y when x increases by 100%
If y is directly proportional to x3, we can write:
y = kx3
where k is the constant of proportionality. To find the percentage increase in y when x increases by 100%, we need to find the new value of y when x is doubled (increased by 100%).
If x is doubled, the new value of x is 2x, so:
y' = k(2x)3
y' = 8kx3
The percentage increase in y is given by:
[(y' - y) / y] x 100%
Substituting the expressions for y and y' and simplifying, we get:
[(8kx3 - kx3) / kx3] x 100%
= 700%
Therefore, when x increases by 100%, y increases by 700%.
If y is directly proportional to x^3, it means that the relationship between y and x can be represented by the equation y = kx^3, where k is a constant.
To find the percentage increase in y when x increases by 100%, we need to first calculate the new value of y when x increases.
Let's assume the initial value of x is x1 and the initial value of y is y1. When x increases by 100%, the new value of x, denoted as x2, will be equal to 2x1.
Using the equation y = kx^3, we can calculate the new value of y, denoted as y2, when x = x2:
y2 = k(x2^3)
= k((2x1)^3)
= 8k(x1^3)
Therefore, the new value of y, y2, is 8 times the initial value of y, y1 when x increases by 100%.
To find the percentage increase in y, we can use the formula:
Percentage increase = ((y2 - y1) / y1) * 100
Substituting the values, we get:
Percentage increase = ((8y1 - y1) / y1) * 100
= (7y1 / y1) * 100
= 700%
So, the percentage increase in y when x increases by 100% is 700%.