1. The quantity Q is proportional to x^3 and inversely proportional to
(y+a)^4 . Find a formula for Q given Q=9 when x=3 and y=0.
2. Also :
Express (32/27) / 18 x (square root of) 3.
in the form 2^a x 3^b
thankkkks!
Hopefully I am reading this correctly, but I would start by factoring the numbers and then separating them.
32 = 2^5
27 = 3^3
18 = 2 * 3^2
sqrt(3) = 3^1/2
So, the original form can be rewritten as:
(32/2) * 1/(3^3*3^2*3^(1/2))
which can be reduced to
16/(3^(11/2)) or 16/(3^5.5)
The exponent of 5.5 is found by adding 3 + 2 + 0.5 since multiplying numbers with the same base, which in this case is 3, and different exponents requires that they be added.
Now, our task is to express it in the form of 2^a * 3^b.
16 = 2^4
So, the fraction looks like:
(2^4)/(3^5.5)
Since 3^5.5 is in the denominator, the exponent becomes negative, and the final result looks like:
2^4 * 3^-5.5 or 2^4 * 3^(-11/2)
I hope this is helpful.
If Q is proportional to x^3 and also Q is proportional to 1/(y+a)^4, then we could set up an equation along the lines of
Q = x^3 / (y+a)^4
We are given initial values:
Q = 9, x = 3 and y = 0.
Substitute these to yield:
9 = 3^3 / (0+a)^4
Reduce: 9 = 27/a^4
Clear the fractions by multiplying both sides by a^4 and dividing by 9 to get:
a^4 = 3
Solve for a: a = 3^(1/4) or 3^0.25
Now, rewrite the original equation
Q = x^3 / (y + 3^0.25)^4
Hello,
"Q is proportional to x^3" means that there exists a number k such that $\displaystyle Q=k*x^3$
"Q is inversely proportional to (y+a)^4" means that there exists a number m such that $\displaystyle Q=\frac{m}{(y+a)^4}$
When $\displaystyle x=3$, we get from the first formula, $\displaystyle 9=Q=k*(3)^3 \implies k=\frac{9}{3^3}=\frac 13$Find a formula for Q given Q=9 when x=3 and y=0.
When $\displaystyle y=0$, we get from the second formula, $\displaystyle 9=Q=\frac{m}{a^4} \implies m=9a^4$
So there are two formulae for Q (by substituting k and m) :
$\displaystyle Q=\frac{x^3}{3}=\frac{9a^4}{(y+a)^4}$
=$\displaystyle \frac{32}{27} \times \frac{1}{18}\times \sqrt{3}$
$\displaystyle = \frac{32}{27 \times 18} \times 3^{\frac{1}{2}}$
$\displaystyle = \frac{2^5}{3^3 \times 2 \times 3^2} \times 3^{\frac{1}{2}}$
$\displaystyle
= \frac{2^4}{3^5} \times 3^{\frac{1}{2}}$
$\displaystyle =2^4 \times 3^{(\frac{1}{2}-5)}$
$\displaystyle =2^4 \times 3^{\frac{-9}{2}}$