$\bf log_{{ a}}(xy)\implies log_{{ a}}(x)+log_{{ a}}(y)\\\\
and\qquad \textit{difference of squares}
\\ \quad \\
(a-b)(a+b) = a^2-b^2\qquad \qquad 
a^2-b^2 = (a-b)(a+b)\\\\
-----------------------------\\\\
ln(a+3)+ln(a-3)=ln(16)\implies ln[(a+3)(a-3)]=ln(16)
\\\\\\
ln[a^2-3^2]=ln(16)\impliedby \textit{removing ln() from both sides}
\\\\\\
a^2-9=16\implies a^2=16+9\implies a^2=25
\\\\\\
a=\pm\sqrt{25}\implies a=\pm 5$

7 0

3 years ago

Find the domain for the particular solution to the differential equation dy dx equals the quotient of 3 times y and x , with ini

Svetach [21]

For this case we have the following difference equation:
$dy / dx = 3xy
$
Applying separable variables we have:
$dy / y = 3xdx
$
Integrating both sides we have:
$\int\ ({1/y}) \, dy = \int\ {3x} \, dx$
$ln (y) = (3/2) x ^ 2 + C
$
applying exponential to both sides:
$exp (ln (y)) = exp ((3/2) x ^ 2 + C)

 y = C * exp ((3/2) x ^ 2)$
For y (1) = 1 we have:
$C = 1 / (exp ((3/2) * 1 ^ 2))

C = 0.2$
Thus, the particular solution is:
$y = 0.2 * exp ((3/2) x ^ 2)
$
Whose domain is all real.
Answer:
y = 0.2 * exp ((3/2) x ^ 2)
Domain: all real numbers

6 0

3 years ago

a ladder leans against the sufe of a house. the angle of elevation of the ladder is 70 degrees when the bottom of the ladder is

nekit [7.7K]

Answer:

≈ 35.1 ft

Step-by-step explanation:

The model is a right triangle with ladder being the hypotenuse and the angle between the ground and the ladder is 70°

Using the cosine ratio, with l being the length of the ladder.

cos70° = $\frac{adjacent}{hypotenuse}$ = $\frac{12}{l}$ ( multiply both sides by l )