1. The area of a square varies directly as the square of its side. If the side is halved, what would be the effect on its area?
2. If y varies directly as the square of x, what will happen to y when x is tripled?
a = ks^2
new area: k(s/2)^2 = ks^2/4 = a/4
y=kx^2
k(3x)^2 = 9kx^2 = 9y
1. If the area of a square varies directly as the square of its side, then we can write the equation as A = k * s^2, where A represents the area, s represents the side length, and k is the constant of variation.
If we halve the side length, let's say s becomes s/2, then the new area, which we'll call A', can be found by substituting the new side length into the equation: A' = k * (s/2)^2 = k * (s^2/4) = (1/4) * (k * s^2) = (1/4) * A.
So, when the side length is halved, the area is reduced to one-fourth (1/4) of its original value.
2. If y varies directly as the square of x, then we can write the equation as y = k * x^2, where y represents the dependent variable, x represents the independent variable, and k is the constant of variation.
When x is tripled, it becomes 3x. We can find the new value of y, which we'll call y', by substituting 3x into the equation: y' = k * (3x)^2 = k * (9x^2) = 9 * (k * x^2) = 9 * y.
So, when x is tripled, y is multiplied by 9.
1. To determine the effect of halving the side of a square on its area, we need to understand the concept of direct variation. In direct variation, two variables are related in such a way that when one variable increases (or decreases) by a certain factor, the other variable also increases (or decreases) by the same factor.
In this case, the area of a square (A) is said to vary directly as the square of its side (s). Mathematically, we can express this relationship as A = k * s^2, where k is the constant of variation.
Now, let's consider what happens when the side (s) is halved. If the original side length is s, then the halved side length would be s/2.
To find the effect on the area, we substitute the halved side length into the equation for the area: A = k * (s/2)^2 = k * (s^2/4).
Therefore, halving the side of the square would result in the area becoming one-fourth (1/4) of the original area. In other words, the area would decrease by 75%.
2. To determine what will happen to y when x is tripled, we need to refer to the concept of direct variation once again.
In this scenario, y varies directly as the square of x. Mathematically, we can express this relationship as y = k * x^2, where k is the constant of variation.
Now, let's consider what happens when x is tripled. If the original value of x is x, then tripling it would result in 3x.
To find the effect on y, we substitute the tripled value of x into the equation for y: y = k * (3x)^2 = k * 9x^2.
Therefore, tripling the value of x results in y becoming nine times (9x) the original value. In other words, y would increase by a factor of 9.