Answer:
[tex] a = 72.39 [/tex]
Step-by-step explanation:
If the variable [tex]a[/tex] is jointly proportional to the cube of [tex]b[/tex] and the square of [tex]c[/tex], we can express this relationship with the equation:
[tex] \Large\boxed{\boxed{a = k \times b^3 \times c^2}} [/tex]
where
- [tex]k[/tex] is the constant of proportionality.
To find the value of [tex]k[/tex], we can use the given information that when [tex]a = 433[/tex], [tex]b = 5[/tex], and [tex]c = 7[/tex].
Substitute the values into the equation:
[tex] 433 = k \times 5^3 \times 7^2 [/tex]
[tex] 433 = k \times 125 \times 49 [/tex]
[tex] 433 = 6125k [/tex]
Now, solve for [tex]k[/tex]:
[tex] k = \dfrac{433}{6125} [/tex]
Now that we have the value of [tex]k[/tex], we can use it to find the value of [tex]a[/tex] when [tex]b = 4[/tex] and [tex]c = 4[/tex]:
[tex] a = \dfrac{433}{6125} \times 4^3 \times 4^2 [/tex]
[tex] a = \dfrac{433}{6125}\times 64 \times 16 [/tex]
[tex] a = \dfrac{433}{6125}\times 1024 [/tex]
[tex] a = 72.39053061 [/tex]
[tex] a = 72.39 \textsf{(in nearest two decimal places)}[/tex]
Therefore, when [tex]b = 4[/tex] and [tex]c = 4[/tex], the value of [tex]a[/tex] is approximately [tex]72.39[/tex].
