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2 years ago

A cube has dimensions of 0.60 meters on each side. what is the volume of the cube in liters?

1 answer:

8 0

The relation of Liters and meters cubed is as follows:

$1L=1dm^3$ , where 1 dm is 10 cm that is 0.1 m.

$(1dm)^3=(0.1m)^3=0.001m^3$

thus: $1L=0.001m^3$ , this means that $1m^3=1000L$

The volume of a cube with side length x is $V=x^3$ .

Thus, the volume of a cube with dimensions 0.60 m is $V=0.6^3=0.216(m^3)=0.216\cdot 1000L=216 L$

Answer: 216L

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F(x, y) = x2 + y2 + 4x − 4y, x2 + y2 ≤ 81 find the extreme values of f on the region described by the inequality.

Whitepunk [10]

Take partial derivatives and set them equal to 0:

$\nabla F(x,y)=\langle2x+4,2y-4\rangle=\mathbf 0$

We find one critical point within the boundary of the disk at $(x,y)=(-2,2)$ . The Hessian matrix for this function is

$\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}2&0\\0&2\end{bmatrix}$

which is positive definite, and incidentally independent of $x$ and $y$ , so $F(x,y)$ attains a minimum $F(-2,2)=-8$ .

Meanwhile, we can parameterize the boundary by $\mathbf r(t)=\langle9\cos t,9\sin t\rangle$ with $0\le t\le2\pi$ , which gives

$F(x,y)=F(x(t),y(t))=f(t)=81\cos^2t+81\sin^2t+36\cos t-36\sin t=81+36(\cos t-\sin t)$

with critical points at

$f'(t)=0\implies 36(-\sin t-\cos t)=0\implies\sin t+\cos t=0\implies t=\dfrac{3\pi}4,\dfrac{7\pi}4$

At these points, we get

$f\left(\dfrac{3\pi}4\right)=81-36\sqrt2\approx30.0883$
$f\left(\dfrac{7\pi}4\right)=81+36\sqrt2\approx131.9117$

so we attain a maximum only when $t=\dfrac{7\pi}4$ , which translates to $(x,y)=\left(\dfrac9{\sqrt2},-\dfrac9{\sqrt2}\right)$ .

3 0

2 years ago

Suppose the weights of fish caught from pier 1 and pier 2 are normally distributed with equal population standard deviations. Th

madreJ [45]

Answer:

H₀: μ₁ = μ₂ vs, Hₐ: μ₁ > μ₂.

Step-by-step explanation:

A two-sample z-test can be performed to determine whether the claim made by the owner of pier 1 is correct or not.

It is provided that the weights of fish caught from pier 1 and pier 2 are normally distributed with equal population standard deviations.

The hypothesis to test whether the average weights of the fish in pier 1 is more than pier 2 is as follows:

H₀: The weights of fish in pier 1 is same as the weights of fish in pier 2, i.e. μ₁ = μ₂.

Hₐ: The weights of fish in pier 1 is greater than the weights of fish in pier 2, i.e. μ₁ > μ₂.

The significance level of the test is:

α = 0.05.

The test is defined as:

$z=\frac{(\bar x_{1}-\bar x_{2})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$

The decision rule for the test is:

If the p-value of the test is less than the significance level of 0.05 then the null hypothesis will be rejected and vice-versa.

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2 years ago

Motorola used the normal distribution to determine the probability of defects and the number of defects expected in a production

Whitepunk [10]

Answer:

32% probability of a defect.

The expected number of defects for a 1000-unit production run is 320.

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 10

Standard deviation = 0.15.

Probability of a defect

Weights less than 9.85 oz or greater than 10.15 oz will be classified as defects.

9.85 = 10 - 0.15

10.15 = 10 + 0.15

By the Empirical Rule, 68% of weights are between 9.85 and 10.15, that is, within the limits. 32% are not within the limits.

So there is a 32% probability of a defect.

Expected number of defects for a 1000-unit production run.

E(X) = 0.32*1000 = 320

The expected number of defects for a 1000-unit production run is 320.

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2 years ago

What system of linear inequalities is shown in the graph?

vesna_86 [32]

U have a dashed line.....it has a y int of 2...it has slope of 1/3....it is shaded below the line....this inequality is : y < 1/3x + 2

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2 years ago

Ramon is having some friends over after a baseball game. Ramon's job is to make vegetable dip. The ingredients for the recipe ar

diamong [38]

Answer:

Butter milk

Step-by-step explanation:

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2 years ago