If y varies directly as the square root of x and y = 47 when x = 49, find y if x = 2401. (Round off your answer to the nearest hundredth.)
y = k√x
That is, y/√x = k, a constant. So, you want y such that
y/√2401 = 47/√49
Well, if y varies directly as the square root of x, we can set up a proportion to solve for y.
We know that when x = 49, y = 47. Let's call this y₁ and x₁.
So we have:
y₁ / x₁^(1/2) = y₂ / x₂^(1/2)
Substituting in the values we have so far:
47 / 49^(1/2) = y₂ / 2401^(1/2)
Simplifying a bit, we get:
47 / 7 = y₂ / 49
Cross multiplying, we have:
7y₂ = 47 * 49
Dividing both sides by 7, we get:
y₂ = 47 * 7 = 329
So when x = 2401, y is approximately 329.
Remember, I'm just a bot, so take my answer with a grain of salt... or maybe a clown nose. 🤡
To find the value of y when x = 2401, we first need to establish the relationship between y and x.
We are given that y varies directly as the square root of x, which can be expressed as:
y = k√x
Where k is the constant of variation.
Using the given information when x = 49 and y = 47, we can substitute these values into the equation to solve for k.
47 = k√49
Taking the square root of 49 results in:
47 = k(7)
To solve for k, we divide both sides of the equation by 7:
47/7 = k
Simplifying, we find:
k = 6.71 (rounded to the nearest hundredth)
Now that we know the value of k, we can substitute it into the equation y = k√x to solve for y when x = 2401:
y = 6.71√2401
Taking the square root of 2401:
y = 6.71(49)
y = 328.79 (rounded to the nearest hundredth)
Therefore, when x = 2401, y ≈ 328.79.
To find the value of y when x = 2401, we need to use the concept of direct variation and the given values for y and x. Let's break down the problem and explain how to get the answer step by step.
1. First, let's understand what it means for y to vary directly as the square root of x. Direct variation means that as one variable (in this case, x) increases or decreases, the other variable (y) also increases or decreases proportionally. In mathematical terms, this can be represented as:
y = k √x
Where k is the constant of variation. The square root sign (√) indicates that y depends on the square root of x.
2. We are given that when x = 49, y = 47. Let's substitute these values into the equation to find the value of k:
47 = k √49
3. Simplify the equation by finding the square root of 49:
47 = k * 7
4. Solve for k by dividing both sides of the equation by 7:
k = 47/7
k = 6.71 (rounded to two decimal places)
5. Now that we have the value of k, we can use it to find y when x = 2401:
y = 6.71 √2401
6. Calculate the square root of 2401:
y = 6.71 * 49
7. Multiply 6.71 by 49:
y = 329.79 (rounded to the nearest hundredth)
Therefore, when x = 2401, y is approximately equal to 329.79.