Y is directly proportional to the square of x. Find percentage increase in value of y when value of x is doubled
y = kx^2
when x is doubled, you have
k(2x)^2 = 4kx^2 = 4y
so, y has grown by 300%
(2x)^2 --> 4Y
4Y is 300% greater than Y.
To solve this, we need to determine the relationship between the variables y and x.
The statement "Y is directly proportional to the square of X" can be mathematically represented as:
Y = k * X^2
Where Y is the dependent variable, X is the independent variable, and k is the constant of proportionality.
Now, we are asked to find the percentage increase in the value of y when the value of x is doubled.
Let's proceed step-by-step:
Step 1: Substitute X with 2X in the equation for Y:
Y_new = k * (2X)^2
= k * 4X^2
= 4kX^2
Step 2: Calculate the percentage increase in Y by finding the difference between Y_new and Y, and dividing it by the original value of Y, then multiplying by 100:
Percentage increase = ((Y_new - Y)/Y) * 100
Step 3: Substitute the equations for Y and Y_new:
Percentage increase = ((4kX^2 - kX^2)/kX^2) * 100
= ((3kX^2)/(kX^2)) * 100
= 3 * 100
= 300%
Therefore, the percentage increase in the value of Y when the value of X is doubled is 300%.
To find the percentage increase in the value of y when the value of x is doubled, we need to understand the concept of direct proportionality.
In this scenario, we are told that y is directly proportional to the square of x. This means that as x increases, y will also increase proportionally, and specifically, the ratio of y to x squared will remain constant.
Mathematically, we can express this relationship as:
y = k * x^2
where k is the constant of proportionality.
Now, let's consider the scenario where the value of x is doubled. This means the new value of x (let's call it x') is 2 times the original value of x. Therefore, we can represent the new value of y (y') as:
y' = k * (2x)^2
Simplifying this expression, we get:
y' = 4k * x^2
Comparing the original value of y (y) and the new value of y (y'), we can determine the percentage increase using the following formula:
Percentage increase = (y' - y) / y * 100%
Plugging in the values, we get:
Percentage increase = (4kx^2 - kx^2) / kx^2 * 100%
Simplifying further, we have:
Percentage increase = (3kx^2) / kx^2 * 100%
Notice that the constant of proportionality (k) and the original value of x^2 cancel out, leaving us with:
Percentage increase = 3 * 100%
Therefore, when the value of x is doubled, the value of y increases by 300%.