y is directly proportional to the cube of x. If x is increased by 50% show that the percentage increase in y is 237.5%
Use law of exponents
(ax)^b = a^b × x^b
y=kx^3
y1=k(1.5x)³=k(1.5^3)x³=3.375kx³=...
To show that the percentage increase in y is 237.5% when x is increased by 50%, we need to use the concept of direct proportionality and apply the given relationship between y and x.
In a direct proportion, when one variable (y) is directly proportional to another variable (x) raised to a power, the proportionality constant (k) is introduced. Mathematically, it can be expressed as:
y = k * x^3
where y is directly proportional to the cube of x.
Now, let's consider two scenarios:
Scenario 1: Initial value of x is x1 and the corresponding value of y is y1.
Scenario 2: Increased value of x by 50% is (x1 + 0.5 * x1) = 1.5 * x1. The corresponding value of y in this case will be denoted as y2.
Substituting x1 and y1 into the direct proportion equation, we have:
y1 = k * x1^3
Substituting 1.5 * x1 and y2 into the direct proportion equation, we have:
y2 = k * (1.5 * x1)^3
To find the percentage increase in y when x is increased by 50%, we can calculate the percentage change using the formula:
Percentage Change = ((New Value - Original Value) / Original Value) * 100
Let's substitute the values into the formula:
Percentage Increase in y = ((y2 - y1) / y1) * 100
Substituting the equations for y1 and y2:
Percentage Increase in y = ((k * (1.5 * x1)^3 - k * x1^3) / (k * x1^3)) * 100
Now, simplify the equation:
Percentage Increase in y = (((1.5)^3 - 1) * (x1^3) / (x1^3)) * 100
= ((3.375 - 1) * 100
= 2.375 * 100
= 237.5%
Therefore, when x is increased by 50%, the percentage increase in y is 237.5%.