Which equation represents a proportional situation?

EXAMPLE 5 If F(x, y, z) = 4y2i + (8xy + 4e4z)j + 16ye4zk, find a function f such that ∇f = F. SOLUTION If there is such a functi

Valentin [98]

If there is such a scalar function f, then

$\dfrac{\partial f}{\partial x}=4y^2$

$\dfrac{\partial f}{\partial y}=8xy+4e^{4z}$

$\dfrac{\partial f}{\partial z}=16ye^{4z}$

Integrate both sides of the first equation with respect to x :

$f(x,y,z)=4xy^2+g(y,z)$

Differentiate both sides with respect to y :

$\dfrac{\partial f}{\partial y}=8xy+4e^{4z}=8xy+\dfrac{\partial g}{\partial y}$

$\implies\dfrac{\partial g}{\partial y}=4e^{4z}$

Integrate both sides with respect to y :

$g(y,z)=4ye^{4z}+h(z)$

Plug this into the equation above with f , then differentiate both sides with respect to z :

$f(x,y,z)=4xy^2+4ye^{4z}+h(z)$

$\dfrac{\partial f}{\partial z}=16ye^{4z}=16ye^{4z}+\dfrac{\mathrm dh}{\mathrm dz}$

$\implies\dfrac{\mathrm dh}{\mathrm dz}=0$

Integrate both sides with respect to z :

$h(z)=C$

So we end up with

$\boxed{f(x,y,z)=4xy^2+4ye^{4z}+C}$

7 0

3 years ago

How does the Counting Principle help when determining the sample space for a probability distribution?

andreyandreev [35.5K]

Answer:

This is also known as the Counting rule.

The Fundamental Counting Principle is used in determining all the possible outcomes and the total possible ways different events can be combined with each other. It is usually done by multiplying all the events together to get the total possible outcome. Doing this also helps in determining the sample space of a probability.

For example there are events a, b and c. The total sample space or possible outcome will be a*b*c.

7 0

3 years ago

Tamara invests $8000 in two different accounts. The first account has a simple interest rate of 3% and the second account has a

Paraphin [41]

Answer:

$3200 at 3%
$4800 at 2%

Step-by-step explanation:

Interest earned is proportional to the interest rate, so if the interest earned is the same, the amounts invested must be inversely proportional to the interest rates. That is, for the 3% and 2% accounts, the ratio of money invested is 2:3.

In other words, 2/5 of the money ($3200) was invested at 3%, and 3/5 of the money ($4800) was invested at 2%.

_____

If you need an equation, you can let x represent the amount invested at the highest rate. Then 8000-x is the amount invested at the lower rate. For the interest in the two accounts to be equal, we have ...

3%·x = 2%·(8000-x) . . . . . the amounts of interest earned are the same

3/2·x = 8000 -x . . . . . . . . divide by 2%

5/2·x = 8000 . . . . . . . . . . . add x and simplify

x = 8000·(2/5) = 3200 . . . multiply by the inverse of the x coefficient

8000-x = 8000 -3200 = 4800 . . . . the amount invested at the lower rate

Tamara invested $3200 in the 3% account and $4800 in the 2% account.

_____

She earned $96 in each account for the year.

8 0

3 years ago

Suppose that scores on the mathematics part of a test for eighth-grade students follow a Normal distribution with standard devia

riadik2000 [5.3K]

Answer:

We need an SRS of scores of at least 153.

Step-by-step explanation:

We have that to find our $\alpha$ level, that is the subtraction of 1 by the confidence interval divided by 2. So:

$\alpha = \frac{1-0.9}{2} = 0.05$

Now, we have to find z in the Ztable as such z has a pvalue of $1-\alpha$ .

So it is z with a pvalue of $1-0.05 = 0.95$ , so $z = 1.645$

Now, find M as such

$M = z*\frac{\sigma}{\sqrt{n}}$

In which $\sigma$ is the standard deviation of the population and n is the size of the sample.

How large an SRS of scores must you choose?

This is at least n, in which n is found when $M = 20, \sigma = 150$ . So

$M = z*\frac{\sigma}{\sqrt{n}}$

$20 = 1.645*\frac{150}{\sqrt{n}}$

$20\sqrt{n} = 1.645*150$

$\sqrt{n} = \frac{1.645*150}{20}$

$\sqrt{n} = 12.3375$

$(\sqrt{n})^{2} = (12.3375)^{2}$

$n = 152.2$

Rounding to the next whole number, 153

We need an SRS of scores of at least 153.

8 0

3 years ago

Sixty-three people participated in a study to see if they were left-brain dominant or right-brain dominant. In the study, the pe

Cloud [144]

The two-way table is attached.

We know that 63 people took the survey, and 22 of them were left-handed. This means that 63-22=41 of them are right handed.

Out of the 63 total, 37 are left brain dominant; this means that 63-37=26 are right brain dominant.

Of the 26 that are right brain dominant, 21 are right handed; this means 26-21=5 are left handed.

Of the 22 left-handed people, 5 are right brain dominant; this means 22-5 = 17 are left brain dominant.

Of the 37 left brain dominant people, 17 are left handed; this means 37-17=20 are right handed.

8 0

3 years ago

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