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VikaD [51]
3 years ago
6

I REALLY REALLY NEED HELP. I WILL MARK BRAINLIEST FOR THE BEST ANSWER!!! Question: What are the dimensions of a box with minimal

surface area that has a volume of 64000 cm^3? AND REMEMBER, PLEASE SHOW ALL WORK NECESSARY. THANKS.
God Bless :)
Mathematics
2 answers:
arlik [135]3 years ago
8 0

Answer:

Step-by-step explanation:

By the arithmetic-geometric mean inequality, surface area of any rectangular parallelpiped satisfies

2(xy+yz+zx)≥2(3)(x2y2z2−−−−−−√3)=6V23.

On the other hand, a cube with side lengths V13 has surface area

2(V13V13+V13V13+V13V13)=6V23.

Hence the cube does indeed have minimum surface area

katrin2010 [14]3 years ago
6 0

9514 1404 393

Answer:

  40 cm × 40 cm × 40 cm

Step-by-step explanation:

Since this is in the high-school math forum, we assume that you're not intended to use calculus to determine the answer. All three dimensions are open for variation, so we'll try to use some logic to determine a reasonable solution.

The dimensions of a box are interchangeable with respect to their effect on surface area. You can see this in the area formula:

  A = 2(LW +LH +WH)

That is, no particular dimension makes more or less contribution to surface area than any other. This suggests that the minimum area will be obtained when each dimension is equal to the others. That is, <em>a cube has the minimum surface area for a given volume</em>.

The volume of a cube in terms of its edge dimension s is ...

  V = s³

So, the edge dimension is ...

  s = ∛V = ∛(64000 cm³) = 40 cm

The dimensions of the box are 40 cm × 40 cm × 40 cm.

_____

<em>Calculus method of Lagrange Multipliers</em>

We want to minimize A=2(LW +WH +LH) subject to the constraint that LWH = 64000. We can write the Lagrangian as ...

  <em>L</em> = 2(LW +WH +LH) +λ(LWH -64000)

We want to set all of the partial derivatives of <em>L</em> to zero.

  d<em>L</em>/dL = 0 = 2(W+H) +λWH

  d<em>L</em>/dW = 0 = 2(L+H) +λLH

  d<em>L</em>/dH = 0 = 2(L+W) +λLW

  d<em>L</em>/dλ = 0 = LWH -64000

Solving the first two equations for λ and setting the results equal, we have ...

  -2(W+H)/(WH) = λ = -2(L+H)/(LH)

Multiplying by H/(-2) gives ...

  1 +H/W = 1 +H/L   ⇒   W = L

Similarly, solving the 2nd and 3rd equations for λ and setting the results equal, we have ...

  -2(L+H)/(LH) = λ = -2(L+W)/(LW)

  1 +L/H = 1 +L/W   ⇒   H = W

So, now we know that L = W = H, which is the assumption we started with in our "logical" answer.

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In the right triangle shown Ac=BC and AB=6 to the square root of 3, how long are each of the legs
Inga [223]

Answer:

6 units

Step-by-step explanation:

I will just assume that you made a typo when typing the question when saying AB is 6√3. Here is the solution if AB = 6√2.

Since it is given that ABC is a right triangle and x labels each of the legs, the triangle is a right isoceles triangle.

Now we can use the right isoceles triangle theorem to solve the problem. This theorem states that if a leg is "x" in a right isoceles triangle, then the hypotenuse is equal to x√2.

Here, the hypotenuse is equal to 6√2. To figure out the legs, you need to solve the equation 6√2 = x√2. It is solved here:

6√2 = x√2 (Divide by √2)

x = 6

The length of the legs are 6 units.

4 0
3 years ago
I kinda got confused on this one <br> 25z - 8z + (-3z)
otez555 [7]

Answer:

The answer is 14z

Step-by-step explanation:

25z - 8z + (-3z)

-8z + (-3z) is the same thing as -8z - 3z, which equals -11z,

25z - 11z = 14z

7 0
3 years ago
Read 2 more answers
NEED ASAP PLEASE HELP ME!!!!<br><br>And explain to ok, But please help
kkurt [141]

Asheville had about 20 more inches of rain and it rained for about 40 more days than Wichita

subtract both rainfall and days for Wichita from the same quantities in Asheville

47.71 - 28.61 = 19.1 , about 20 inches

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both quantities rounded to the nearest 10




3 0
3 years ago
Read 2 more answers
Can someone tell me how to list these numbers from least to greatest. -4, 6.2, 18 1/2, -5,9, 21, -1/4, and 1.75. Thanks
Fiesta28 [93]
I think it is -5,-4,-1/4, 1.75, 6.2, 9, 18 1/2, 21
4 0
3 years ago
An automobile manufacturer who wishes to advertise that one of its models achieves 30 mpg (miles per gallon) decides to carry ou
Katyanochek1 [597]

Answer:

t=\frac{29.35-30}{\frac{1.365}{\sqrt{6}}}=-1.17  

p_v =P(t_{(5)}  

If we compare the p value and the significance level given \alpha=0.05 we see that p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis.  

We can say that at 5% of significance the true mean is not significantly less than 30 so then the claim that true average fuel efficiency is (at least) 30 mpg makes sense

Step-by-step explanation:

Data given and notation  

The mean and sample deviation can be calculated from the following formulas:

\bar X =\frac{\sum_{i=1}^n x_i}{n}

s=\sqrt{\frac{\sum_{i=1}^n (x_i -\bar X)}{n-1}}

\bar X=29.35 represent the sample mean  

s=1.365 represent the sample standard deviation  

n=6 sample size  

\mu_o =30 represent the value that we want to test  

\alpha=0.05 represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

p_v represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the true mean is lower than 30 or no, the system of hypothesis are :  

Null hypothesis:\mu \geq 30  

Alternative hypothesis:\mu < 30  

Since we don't know the population deviation, is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}} (1)  

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic  

We can replace in formula (1) the info given like this:  

t=\frac{29.35-30}{\frac{1.365}{\sqrt{6}}}=-1.17  

P-value  

We need to calculate the degrees of freedom first given by:  

df=n-1=6-1=5  

Since is a one-side left tailed test the p value would given by:  

p_v =P(t_{(5)}  

Conclusion  

If we compare the p value and the significance level given \alpha=0.05 we see that p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis.  

We can say that at 5% of significance the true mean is not significantly less than 30 so then the claim that true average fuel efficiency is (at least) 30 mpg makes sense

5 0
3 years ago
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