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3 years ago

Solve the logarithmic equation. When necessary, round answer to the nearest hundredth. log 3(x + 4) - log 3(x + 2) = log 327

Mathematics

1 answer:

alexandr1967 [171]3 years ago

8 0

Answer:

$x = -\dfrac{25}{13}$

Step-by-step explanation:

$\log_3{x + 4} - \log_3{x + 2} = \log_3 {27}$

$\log_3 \dfrac{x + 4}{x + 2} = \log_3 {27}$

$\dfrac{x + 4}{x + 2} = 27$

$x + 4 = 27(x + 2)$

$x + 4 = 27x + 54$

$-26x = 50$

$x = -\dfrac{50}{26}$

$x = -\dfrac{25}{13}$

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3. What kind of triangle is represented by the triangle shown below?

krek1111 [17]

Answer:

Isosceles

Step-by-step explanation:

An Isosceles triangle has an acute angle (less than 45°), and this photo shows a triangle with two matching sides with one side being longer.

A right triangle would have a 90° angle which this photo doesn't have

An Equilateral triangle has all 60° sides, since it doesn't have all matching sides, it cannot be the answer.

A Scalene triangle is similar to this photo except it has 30° 60° and 90° angles, which this photo doesn't have.

I hope this helps!

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1 year ago

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Nimfa-mama [501]

Step-by-step explanation:

From both the cases we can say that √n is either a natural number or an irrational number. Hence the correct option is (d)Either a natural number or an irrational number. Note: Before solving this type of question we should have proper knowledge of natural numbers, rational numbers and irrational numbers.

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2 years ago

What pair of points have a slope of -3/5

sergey [27]

Answer:

(5,0) and (0,3) have a slope of -3/5

Step-by-step explanation:

Here, we want to state a pair of points that have a slope of -3/5

Mathematically, we can have the slope of a straight line calculated as;

m = y2-y1/(x2-x1)

This could give points like (5,0) and (0,3)

3 0

2 years ago

Find all the solutions for the equation:

Contact [7]

$2y^2\,\mathrm dx-(x+y)^2\,\mathrm dy=0$

Divide both sides by $x^2\,\mathrm dx$ to get

$2\left(\dfrac yx\right)^2-\left(1+\dfrac yx\right)^2\dfrac{\mathrm dy}{\mathrm dx}=0$

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2\left(\frac yx\right)^2}{\left(1+\frac yx\right)^2}$

Substitute $v(x)=\dfrac{y(x)}x$ , so that $\dfrac{\mathrm dv(x)}{\mathrm dx}=\dfrac{x\frac{\mathrm dy(x)}{\mathrm dx}-y(x)}{x^2}$ . Then

$x\dfrac{\mathrm dv}{\mathrm dx}+v=\dfrac{2v^2}{(1+v)^2}$

$x\dfrac{\mathrm dv}{\mathrm dx}=\dfrac{2v^2-v(1+v)^2}{(1+v)^2}$

$x\dfrac{\mathrm dv}{\mathrm dx}=-\dfrac{v(1+v^2)}{(1+v)^2}$

The remaining ODE is separable. Separating the variables gives

$\dfrac{(1+v)^2}{v(1+v^2)}\,\mathrm dv=-\dfrac{\mathrm dx}x$

Integrate both sides. On the left, split up the integrand into partial fractions.

$\dfrac{(1+v)^2}{v(1+v^2)}=\dfrac{v^2+2v+1}{v(v^2+1)}=\dfrac av+\dfrac{bv+c}{v^2+1}$

$\implies v^2+2v+1=a(v^2+1)+(bv+c)v$

$\implies v^2+2v+1=(a+b)v^2+cv+a$

$\implies a=1,b=0,c=2$

Then

$\displaystyle\int\frac{(1+v)^2}{v(1+v^2)}\,\mathrm dv=\int\left(\frac1v+\frac2{v^2+1}\right)\,\mathrm dv=\ln|v|+2\tan^{-1}v$

On the right, we have

$\displaystyle-\int\frac{\mathrm dx}x=-\ln|x|+C$

Solving for $v(x)$ explicitly is unlikely to succeed, so we leave the solution in implicit form,

$\ln|v(x)|+2\tan^{-1}v(x)=-\ln|x|+C$

and finally solve in terms of $y(x)$ by replacing $v(x)=\dfrac{y(x)}x$ :

$\ln\left|\frac{y(x)}x\right|+2\tan^{-1}\dfrac{y(x)}x=-\ln|x|+C$

$\ln|y(x)|-\ln|x|+2\tan^{-1}\dfrac{y(x)}x=-\ln|x|+C$

$\boxed{\ln|y(x)|+2\tan^{-1}\dfrac{y(x)}x=C}$

7 0

3 years ago

A swimming pool is to be drained. The pool is shaped like a rectangular prism with length 30 feet, wide 18 ft, and depth 4ft. Su

Gala2k [10]

Answer: it will take 9 hours to empty the pool.

Step-by-step explanation:

The pool is shaped like a rectangular prism with length 30 feet, wide 18 ft, and depth 4ft. It means that when the pool is full, its volume is

30 × 18 × 4 = 2160 ft³

If water is pumped out of the pool at a rate of 216ft3 per hour, then the rate at which the water in the pool is decreasing is in arithmetic progression. The formula for determining the nth term of an arithmetic sequence is expressed as

Tn = a + d(n - 1)

Where

a represents the first term of the sequence(initial amount of water in the pool when completely full).

d represents the common difference(rate at which it is being pumped out)

n represents the number of terms(hours) in the sequence.

From the information given,

a = 2160 degrees

d = - 216 ft3

Tn = 0(the final volume would be zero)

We want to determine the number of terms(hours) for which Tn would be zero. Therefore,

0 = 2160 - 216 (n - 1)

2160 = 216(n - 1) = 216n + 216

216n = 2160 - 216

216n = 1944

n = 1944/216

n = 9

8 0

3 years ago