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pav-90 [236]
3 years ago
14

Felipe made sand art bottles to sell at his school's craft fair. He wanted to use 9 different colors of sand. So, he bought 1 po

und of each color. Then, he filled as many 6-ounce bottles as he could. How many sand art bottles did Felipe make?
Mathematics
1 answer:
Dennis_Churaev [7]3 years ago
8 0

Answer:  Felipe made 24 sand art bottles.

Step-by-step explanation:

Given: Total different colors of sand = 9

Weight of each color sand = 1 pound

Then, the weight of all 9 colors would be  9 × 1 = 9 pounds

Since 1 pound = 16 ounces

Then, 9 pounds = 9 × 16 = 144 ounces

capacity of each bottle = 6 ounce

Then, the number of bottles required = ( 144 ounces) ÷ 6

= 24

Hence,  Felipe made 24 sand art bottles.

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Can anyone explain why this two is being added before the integral? Does it have something to do with integral symmetry?
PIT_PIT [208]

Yes it's exactly due to symmetry. Specifically, symmetry about the x axis.

Simply writing \displaystyle \int_{-8}^{-4}\left[\sqrt{x+8} \ \right]dx, without the 2 out front, will only get you the area of the portion shown in red (see diagram below).

The blue region has equal area of the red region due to symmetry.

So,

\displaystyle 2\int_{-8}^{-4}\left[\sqrt{x+8} \ \right]dx

represents the red and blue regions combined.

The portion in green is \displaystyle \int_{-4}^{8}\left[\sqrt{x+8} - \frac{x}{2} \ \right]dx which is the integral of the difference of the upper and lower curves over the interval -4 \le x \le 8

---------------

In all honesty, it's probably easier to integrate with respect to y since the given functions are in terms of y initially. Also, there isn't a junction point in which the curves swap places in terms of which one is larger. However, it doesn't hurt to have practice in integrating with respect to x.

If you're curious about what the y integral looks like, then it would be

\displaystyle \int_{-2}^{4} \bigg[ (2y) - (y^2-8)\bigg] dy

You can use a tool like WolframAlpha to check that both integral expressions result in 36 to help confirm that they represent the same overall area (just in different ways of course).

5 0
2 years ago
Read 2 more answers
25x power square - 36 y power square​
finlep [7]

\implies {\blue {\boxed {\boxed {\purple {\sf {  (5x + 6y)(5x - 6y) }}}}}}

\large\mathfrak{{\pmb{\underline{\orange{Step-by-step\:explanation}}{\orange{:}}}}}

={25x}^{2}  - 36 {y}^{2}

=  ({5 \times 5)x}^{2}  -  (6 \times 6){y}^{2}

=  ({5x})^{2}  -(  {6y})^{2}

= (5x + 6y)(5x - 6y)

<h3><u>Note</u>:</h3>

{a}^{2}  -  {b}^{2}  = (a + b)(a - b)

\red{\large\qquad \qquad \underline{ \pmb{{ \mathbb{ \maltese \: \: Mystique35♛}}}}}

3 0
3 years ago
What is the vertex of the graph of the function below y=x^2-8x+12
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That is the answer Y=x^-8x12, xER
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Select all the equations that have the same solution as the equation 3x – 12 = 24
Stolb23 [73]
Your answer will be D and E
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Statistics Question. Use the Desmos graphing calculator to find the least-squares regression line for the dataset in the table:
Soloha48 [4]

Given:

The table of values.

To find:

The least-squares regression line for the data set in the table by using the desmos graphing calculator.

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The general form of least-squares regression line is:

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