Certainly! Let's solve the question step by step.
### Given:
- [tex]\( a = 21 \)[/tex] when [tex]\( b = 5 \)[/tex] and [tex]\( c = 36 \)[/tex]
- [tex]\( a \)[/tex] varies jointly as [tex]\( b \)[/tex] and the square root of [tex]\( c \)[/tex], i.e., [tex]\( a = k \cdot b \cdot \sqrt{c} \)[/tex]
### Step-by-step solution:
#### (a) Finding [tex]\( k \)[/tex], the constant of variation:
We start with the relationship [tex]\( a = k \cdot b \cdot \sqrt{c} \)[/tex].
Given values:
- [tex]\( a = 21 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- [tex]\( c = 36 \)[/tex]
We substitute these into the equation to solve for [tex]\( k \)[/tex]:
[tex]\[ 21 = k \cdot 5 \cdot \sqrt{36} \][/tex]
Since [tex]\( \sqrt{36} = 6 \)[/tex]:
[tex]\[ 21 = k \cdot 5 \cdot 6 \][/tex]
[tex]\[ 21 = k \cdot 30 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{21}{30} \][/tex]
[tex]\[ k = 0.7 \][/tex]
So, the constant of variation [tex]\( k \)[/tex] is [tex]\( \boxed{0.7} \)[/tex].
#### (b) Finding [tex]\( a \)[/tex] when [tex]\( b = 9 \)[/tex] and [tex]\( c = 100 \)[/tex]:
Using the previously found value of [tex]\( k \)[/tex] and the general formula [tex]\( a = k \cdot b \cdot \sqrt{c} \)[/tex]:
Given values:
- [tex]\( b = 9 \)[/tex]
- [tex]\( c = 100 \)[/tex]
Substitute these into the equation:
[tex]\[ a = 0.7 \cdot 9 \cdot \sqrt{100} \][/tex]
Since [tex]\( \sqrt{100} = 10 \)[/tex]:
[tex]\[ a = 0.7 \cdot 9 \cdot 10 \][/tex]
[tex]\[ a = 0.7 \cdot 90 \][/tex]
[tex]\[ a = 63 \][/tex]
So, [tex]\( a \)[/tex] when [tex]\( b = 9 \)[/tex] and [tex]\( c = 100 \)[/tex] is [tex]\( \boxed{63} \)[/tex].
#### (c) Finding [tex]\( c \)[/tex] when [tex]\( a = 70 \)[/tex] and [tex]\( b = 25 \)[/tex]:
Using the constant [tex]\( k \)[/tex] again, and the general formula [tex]\( a = k \cdot b \cdot \sqrt{c} \)[/tex]:
Given values:
- [tex]\( a = 70 \)[/tex]
- [tex]\( b = 25 \)[/tex]
We need to solve for [tex]\( c \)[/tex]. Substitute the given values into the equation:
[tex]\[ 70 = 0.7 \cdot 25 \cdot \sqrt{c} \][/tex]
Solving for [tex]\( \sqrt{c} \)[/tex]:
[tex]\[ 70 = 17.5 \cdot \sqrt{c} \][/tex]
[tex]\[ \sqrt{c} = \frac{70}{17.5} \][/tex]
[tex]\[ \sqrt{c} = 4 \][/tex]
Squaring both sides:
[tex]\[ c = 4^2 \][/tex]
[tex]\[ c = 16 \][/tex]
So, [tex]\( c \)[/tex] when [tex]\( a = 70 \)[/tex] and [tex]\( b = 25 \)[/tex] is [tex]\( \boxed{16} \)[/tex].
Hence, our answers are:
- [tex]\( k = \boxed{0.7} \)[/tex]
- [tex]\( a = \boxed{63} \)[/tex] when [tex]\( b = 9 \)[/tex] and [tex]\( c = 100 \)[/tex]
- [tex]\( c = \boxed{16} \)[/tex] when [tex]\( a = 70 \)[/tex] and [tex]\( b = 25 \)[/tex]
