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mr Goodwill [35]
3 years ago
11

A box (with no top) is to be constructed from a piece of cardboard of sides a and b by cutting out squares of length h from the

corners and folding up the sides. find the value of h that maximizes the volume of the box if a = 8 and b = 9.
Mathematics
1 answer:
melisa1 [442]3 years ago
3 0
<span>The volume of the box is h(8-2h)(9-2h). Differentiating this gives dV/dh = 12h^2-68h+72. The derivative is zero when h = (17+sqrt(73))/6 or (17-sqrt(73))/6. The second derivative is positive for the first value, so this gives a minimum volume. The second derivative is negative for the second value, so this gives the maximum volume. So the value of h that maximizes the volume of the box is (17-sqrt(73))/6, which is approximately 1.41.</span>
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SHOW ALL WORK BUT KEE IT SIMPLE AND EASY TO UNDERSTAND:)
7nadin3 [17]
To do this problem, you need to use a process called completing the square. Let me explain:  

To complete the square on the function f(x) = x² + 8x +13, first group the first two terms in (  )  and leave some space at the end as follows:
f(x) = (x² + 8x          ) + 13  Now our next step is to fill in the space and adjust our expression on the right hand side of the function.  To do this, we take half of the middle number 8 and then square it:  so 4² = 16 and we fill in our space inside the ( ) with this value 16;   
f(x) = (x² + 8x + 16) + 13  now what we have done is to increase the overall value of our expression on the right by 16, but we want the overall value to remain the same.  To fix this we simply need to subtract 16 at the end like this:  f(x) = (x² + 8x + 16) + 13 -16   we can simplify and get the following.
f(x) = (x² + 8x + 16) - 3   At this point we're almost done.. All we need to do now is to rewrite the what is in the parentheses in a slightly different form.  Here is what it will look like:  f(x) = (x + 4)² - 3  notice all I did was take the sum of the square root of x² and the square root of 16 originally in the (  ) to get then new expression inside the ( ) and then square that  ( )²

Now this is a nice form to have because you can get the vertex straight from this form.. IN FACT this is called vertex form or (h,k) form for short.  In general the form is f(x) = a(x - h)² + k  don't worry about the 'a'  for now.. you might see that in our case it is just 1 and will not effect our equation.  You only have to consider this if the original leading coefficient of the quadratic is not 1 to begin with...

So you can see that our vertex is (-4,-3)  
Hope this is helpful, but if you have questions let me know.
5 0
3 years ago
A parabola has its vertex at (2, 1) and crosses the x-axis at (1,0) anu al l
Y_Kistochka [10]

Answer:

A

Step-by-step explanation:

The equation of a parabola in vertex form is

y = a(x - h)² + k

where (h, k) are the coordinates of the vertex and a is a multiplier

Here (h, k) = (2, 1), thus

y = a(x - 2)² + 1

To find a substitute (1, 0) into the equation

0 = a(1 - 2)² + 1

0 = a + 1 ( subtract 1 from both sides )

a = - 1

Hence

y = - (x - 2)² + 1 or

y = 1 - (x - 2)² → A

7 0
3 years ago
Find p if 45 :72=72 40:p​
leonid [27]
Here

45 :72 = 40:p
Or 45/72=40/p
Or 45p = 72*40

Or p = 2880/45

Or p= 64

Thank you
Ayanas
4 0
3 years ago
An employer provides two payment options for employees. Option A: Receive $200 the first week. Receive an additional $50 for eac
elena-14-01-66 [18.8K]

Based on the information, it can be inferred that payment option A means a payment of $450, while option B means a payment of $300.

<h3>How much money does each payment option receive?</h3>

The payment options contemplated by the employer are distributed as follows:

Option A

  • First week: $200
  • Second week: $200 + $50 = $250
  • Third week: $250 + $50 = $300
  • Fourth week: $300 + $50 = $350
  • Fifth week: $350 + $50 = $400
  • Sixth week: $400 + $50 = $450

Option B

  • First week: $200
  • Second week: $220
  • Third week: $240
  • Fourth week: $260
  • Fifth week: $280
  • Sixth week: $300

Note: This question is incomplete because there is some information missing. Here is the missing information:

Complete the tables to show how much money would be received for both payment options, each week, for 6 weeks.

Learn more about payment options in: brainly.com/question/18524042

#SPJ1

7 0
2 years ago
The amount of lateral expansion (mils) was determined for a sample of n = 8 pulsed-power gas metal arc welds used in LNG ship co
Alona [7]

Answer:

95% Confidence interval for the variance:

3.6511\leq \sigma^2\leq 34.5972

95% Confidence interval for the standard deviation:

1.9108\leq \sigma \leq 5.8819

Step-by-step explanation:

We have to calculate a 95% confidence interval for the standard deviation σ and the variance σ².

The sample, of size n=8, has a standard deviation of s=2.89 miles.

Then, the variance of the sample is

s^2=2.89^2=8.3521

The confidence interval for the variance is:

\dfrac{ (n - 1) s^2}{ \chi_{\alpha/2}^2} \leq \sigma^2 \leq \dfrac{ (n - 1) s^2}{\chi_{1-\alpha/2}^2}

The critical values for the Chi-square distribution for a 95% confidence (α=0.05) interval are:

\chi_{0.025}=1.6899\\\\\chi_{0.975}=16.0128

Then, the confidence interval can be calculated as:

\dfrac{ (8 - 1) 8.3521}{ 16.0128} \leq \sigma^2 \leq \dfrac{ (8 - 1) 8.3521}{1.6899}\\\\\\3.6511\leq \sigma^2\leq 34.5972

If we calculate the square root for each bound we will have the confidence interval for the standard deviation:

\sqrt{3.6511}\leq \sigma\leq \sqrt{34.5972}\\\\\\1.9108\leq \sigma \leq 5.8819

6 0
3 years ago
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