If a varies jointly as b and the square root of c, and a = 21 when b = 5 and c = 36, find a when b = 9 and c = 225.

To find the value of a when b = 9 and c = 225, we need to determine the constant of variation.

We are given that a varies jointly as b and the square root of c, which can be expressed as:

a = k * b * sqrt(c)

where k is the constant of variation.

Using the first data point, when a = 21, b = 5, and c = 36, we can substitute these values into the equation to find the value of k:

21 = k * 5 * sqrt(36)

First, let's simplify the square root of 36:

21 = k * 5 * 6

Now, we can solve for k by dividing both sides by 5 * 6:

21 / (5 * 6) = k

k = 0.7

Now that we have the value of k, we can use it to find a when b = 9 and c = 225:

a = k * b * sqrt(c)

Replacing b and c with their given values:

a = 0.7 * 9 * sqrt(225)

Before proceeding, let's simplify the square root of 225:

sqrt(225) = 15

Now, we can substitute this value into the equation:

a = 0.7 * 9 * 15

Performing the calculation:

a = 94.5

Therefore, when b = 9 and c = 225, the value of a is 94.5.

To solve this problem, we will follow these steps:

1. Understand the given information:
- We are given that a varies jointly as b and the square root of c.
- We are also given specific values of a, b, and c: a = 21 when b = 5 and c = 36.

2. Formulate the mathematical equation for the given situation:
We can express the joint variation relationship as:
a = k * b * √c
where k is the constant of variation.

3. Determine the constant of variation (k):
To find k, we substitute the given values into the equation and solve for k:
21 = k * 5 * √36
Simplifying this equation, we get:
21 = 5k * 6
Divide both sides of the equation by 30 to isolate k:
21/30 = k
Simplifying further, we have:
7/10 = k

Therefore, the constant of variation (k) is 7/10.

4. Use the constant of variation to find the value of a when b = 9 and c = 225:
Now that we know k, we can substitute the given values of b and c into the equation and solve for a:
a = (7/10) * 9 * √225
Simplifying this equation, we get:
a = (7/10) * 9 * 15
Evaluate the expression to find a:
a = 7 * 9/10 * 15
Multiplying these numbers together, we get:
a = (63/10) * 15
Simplifying further, we have:
a = 945/10
a = 94.5

Therefore, when b = 9 and c = 225, the value of a is 94.5.

1 = k5sqrt(36)

21 = 30k

k = 21/30

a = (21/30)*b sqrt(c)

a = (21/30) *9 sqrt(225)

a = 189/2

a = kbsqrt(c)

21 = k5sqrt936)

21 = 30k

k = 21/30

a = (21/30)*b sqrt(c)

a = (21/30) *9 8sqrt(2250

a = 189/2