It is given that y is directly proportional to x^3. Find the percentage increase in y when x decreases by 50%
Since we know that y is directly proportional to x^3, we can write:
y = kx^3
where k is a constant of proportionality.
Now, let's consider what happens when x decreases by 50%. This means that the new value of x, call it x', is equal to:
x' = 0.5x
To find the corresponding value of y', we substitute x' into our equation for y:
y' = k(x')^3
y' = k(0.5x)^3
y' = 0.125kx^3
So, the new value of y is 0.125 times the original value of y. This represents a decrease of 87.5% in y.
However, we are asked for the percentage increase in y. To find this, we need to use the formula:
% increase = (new value - old value) / old value x 100
Plugging in the values we found, we get:
% increase = ((0.125y - y) / y) x 100
% increase = (-0.875) x 100
% increase = -87.5
So, the percentage increase in y when x decreases by 50% is -87.5%, which represents a decrease in y.
To find the percentage increase in y when x decreases by 50%, we first need to understand the concept of direct proportionality.
In a direct proportion, when one variable (x) increases or decreases, the other variable (y) also increases or decreases proportionally. The equation for direct proportionality is given by y = k * x^n, where k is the constant of proportionality and n is the power to which x is raised.
In this case, it is given that y is directly proportional to x^3, so we can write the equation as y = k * x^3.
Now, let's analyze the situation where x decreases by 50%. If the initial value of x is denoted as x1, the new value of x (after decreasing by 50%) will be x2 = 0.5 * x1.
To find the percentage increase in y, we need to compare the change in y (Δy) to the original value of y.
Let's calculate:
1. Initial value of y: y1 = k * x1^3
2. Final value of y: y2 = k * (0.5 * x1)^3
Now, to find the percentage increase, we can use the formula:
Percentage Increase = (Δy / y1) * 100
Let's substitute the values:
Δy = y2 - y1 = k * (0.5 * x1)^3 - k * x1^3
= k * (0.125 * x1^3) - k * x1^3
= k * (0.125 - 1) * x1^3
= k * (-0.875) * x1^3
Now, we can calculate the percentage increase:
Percentage Increase = (Δy / y1) * 100
= (k * (-0.875) * x1^3) / (k * x1^3) * 100
= (-0.875) * 100
= -87.5%
Therefore, when x decreases by 50%, y decreases by 87.5%.