find the equation that meets the given conditions: a varies directly as the square of r and a=9 pie when r=3
a = kr^2
9pi = k(3^2) = 9k
pi = k
a = pi r^2
Well, well, well, it seems like we have a problem involving direct variation. I can smell the pie in the air! So, we have "a" varying directly as the square of "r" and when "r" is equal to 3, "a" turns into a scrumptious 9 pie. To find the equation, we can set up a proportion:
a/r^2 = k
where "k" is the constant of variation. Now, let's use the given information to solve for "k". We know that when "r" equals 3, "a" equals 9 pie. So, let's plug in these values:
9 pie / (3)^2 = k
And after a little mathematical magic, we get:
k = pie / 1
Therefore, the equation that meets the given conditions is:
a = (pie / 1) * r^2
Or, in a more delicious way:
a = pie * r^2
You've got your equation! Enjoy the pie, my friend!
To find the equation that meets the given conditions, we know that a varies directly as the square of r. In other words, a is directly proportional to r^2.
Let's first express this proportionality relationship as an equation. We can introduce a constant of proportionality, k, to represent the relationship:
a = k * r^2
Next, we will use the provided information that when r = 3, a = 9π to determine the value of the constant k.
Substituting the values into the equation, we have:
9π = k * 3^2
9π = 9k
Now, we solve for k by dividing both sides of the equation by 9:
k = (9π) / 9
Simplifying,
k = π
Now that we have the value of k, we can substitute it back into the equation we derived earlier:
a = π * r^2
This is the equation that meets the given conditions, where a varies directly as the square of r and a = 9π when r = 3.
To find the equation that represents the relationship between "a" and "r," we first need to understand what it means for "a" to vary directly as the square of "r." When two variables vary directly, it means that as one variable changes, the other changes in direct proportion to it. In this case, "a" varies directly with the square of "r," implying that if "r" is multiplied by a factor, "a" will be multiplied by the square of that factor.
Let's break down the problem step by step:
1. Start with the direct variation equation: a = k * r^2, where "k" is the constant of variation.
2. We are given that a = 9π when r = 3. Substituting these values into the equation:
9π = k * 3^2
9π = k * 9
3. Now, solve for "k" by dividing both sides of the equation by 9:
π = k
4. Therefore, the constant of variation "k" is equal to π.
5. Substitute the value of "k" into the original equation:
a = π * r^2
So, the equation that represents the relationship between "a" and "r" is a = π * r^2.