How can you tell whether a rational expression is in simplest form? Include an example with your explanation.

area of a circle = pi r^2

radius = 18

 so R62 = 18^2 = 324

 since you want the answer in pi it would be 324PI

 so the answer is 324

The two numbers with a product equal to 192 such that their sum is minimized are:

A = √192 and B = √192

How to find the two numbers?

Let's define A and B as our two numbers, we know that their product must be equal to 192, then we have:

A*B = 192.

Now, the sum of these two numbers is:

A + B.

And we want to minimize this, to do it, we need to use the first equation to rewrite one variable in terms of the other. For example, if we isolate A, we get:

A = 192/B

Replacing this in the sum, we get:

192/B + B

To minimize this, we need to find the values of B that make 0 the differentiation of the above expression.

The differentiation is:

-192/B^2 + 1

Then we need to solve:

-192/B^2 + 1 = 0

192/B^2 = 1

192 = B^2

√192 = B

To get the value of A, we use:

A = 192/B = 192/√192  = √192

Then we can conclude that the two positive numbers such that their product is 192, and their sum is minimized, is:

A = √192 and B = √192.

If you want to learn more about minimization, you can read:

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Answer:

[tex](-2,-2)[/tex].

Step-by-step explanation:

Let us assume that coordinates of other endpoint are [tex](x_1,y_1)[/tex]

We have been given that the midpoint of a segment is (4,3) and one endpoint is (10,8). We are asked to find the coordinates of the other endpoint.  

We will use midpoint formula to solve our given problem.

[tex]x\text{-coordinate of midpoint}=\frac{x_1+x_2}{2}[/tex]

[tex]y\text{-coordinate of midpoint}=\frac{y_1+y_2}{2}[/tex]

Upon using our given information, we will get:

[tex]4=\frac{x_1+10}{2}[/tex]

[tex]4\cdot 2=\frac{x_1+10}{2}\cdot 2[/tex]

[tex]8=x_1+10[/tex]

[tex]8-10=x_1+10-10[/tex]

[tex]x_1=-2[/tex]

Similarly, we will find y-coordinate.

[tex]3=\frac{y_1+8}{2}[/tex]

[tex]3\cdot 2=\frac{y_1+8}{2}\cdot 2[/tex]

[tex]6=y_1+8[/tex]

[tex]6-8=y_1+8-8[/tex]

[tex]y_1=-2[/tex]

Therefore, the coordinates of other endpoint would be [tex](-2,-2)[/tex].

Final answer:

If one root of a quadratic equation with rational coefficients is rational, the other root must be rational too because the sum and product of the roots are related to the coefficients, which are also rational.

Explanation:

To prove that if one solution for a quadratic equation of the form x^2 + bx + c = 0 is rational, then the other solution is also rational, we can use the quadratic formula and properties of rational numbers. If the quadratic equation has rational coefficients and one rational solution, then the sum and product of the roots must also be rational. This is because a quadratic equation with roots r and s can be factored as (x - r)(x - s) = 0, which expands to x^2 - (r + s)x + rs = 0. Matching coefficients, we see that - (r + s) = b and rs = c. Since b and c are rational, r + s and rs must be rational as well.

Given that we have one rational root, let's say r, the sum of the roots r + s is rational, so s must also be rational because the difference of two rational numbers is rational. Hence, if one root of a quadratic equation with rational coefficients is rational, the other root must be rational as well.

In standard fantasy football, each team incorporates a variety of positions including a Quarterback, Running Backs, Wide Receivers, a Tight End, a Flex, Defense/Special Teams, a Kicker and bench spots.

In standard fantasy football, each team is composed of a variety of positions: 1 Quarterback (QB), 2 Running Backs (RB), 2 Wide Receivers (WR), 1 Tight End (TE), 1 Flex (which can be a RB, WR, or TE), 1 Defense/Special Teams (DST), and 1 Kicker (K). In most cases, you also have bench spots for subs or injured players, usually 6-7. These numbers can vary depending on the rules of your fantasy league.

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This question involves the concept of binomial probability in statistics. The scenarios involve calculating the probability of a certain number of successes (cashew preference) among a set number of trials (12 adults). It's recommended to use a calculator or software tool to perform the calculations.

The subject of the query is known as binomial probability which is part of statistics in mathematics. Here, we need to find the probability of 'success' (in this case, the preference of cashews) exactly a set number of times when conducting a set number of trials (12 adults).

It's also important to note, the results in these examples would be more accurate if you use a calculator or software like Excel to handle the computations.

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The probability of exactly 3 cashews in a random sample of 12 adults is approximately 19.4%.

The probability of at least 4 cashews is approximately 15.0%.

The probability of at most 2 cashews is approximately 72.0%.

These probabilities were calculated using the binomial probability formula and considering the complementary event for cases with "at least" or "at most" conditions. Technology like calculators or statistical software can be helpful in performing these calculations efficiently.

Determining the Probability of Nut Preference in a Random Sample

In this scenario, we're analyzing the probability of specific outcomes when randomly selecting 12 adults and inquiring about their favorite nut, knowing that 43% prefer cashews. Here's a breakdown of the requested probabilities:

(a) Exactly Three Cashew Preferences:

Imagine having 12 slots to fill with "cashew" or "other." We need 3 "cashew" slots and 9 "other" slots. Using the binomial probability formula, the probability of this specific arrangement is:

P(3 cashews in 12 trials) = (12 choose 3) * (0.43)^3 * (0.57)^9 ≈ 0.194

(b) At Least Four Cashew Preferences:

This includes scenarios with 4, 5, 6, 7, 8, 9, 10, 11, or all 12 adults preferring cashews. We can calculate the probability for each case and sum them up, but a simpler approach is to find the probability of the opposite event (fewer than 4 cashews) and subtract it from 1:

P(at least 4 cashews) = 1 - P(fewer than 4 cashews)

P(at least 4 cashews) ≈ 1 - (0.194 + 0.237 + 0.184 + 0.115 + 0.063 + 0.034 + 0.015 + 0.006 + 0.002) ≈ 0.150

(c) At Most Two Cashew Preferences:

This includes scenarios with 0, 1, or 2 adults preferring cashews. Similar to (b), we can calculate the probability for each case and sum them up:

P(at most 2 cashews) = P(0 cashews) + P(1 cashew) + P(2 cashews)

P(at most 2 cashews) ≈ 0.237 + 0.299 + 0.184 ≈ 0.720

Answer:            ≈ 75,779

Step-by-step explanation:

84/4=21

 now take the 2 even numbers below 21 and the 2 even numbers above 21

18 +20 + 22 +24 = 84

 the numbers are 18, 20, 22 & 24

the value of a is

[tex]\( a = \frac{9845}{x - 10} \)[/tex].

To find a, we can use polynomial long division or synthetic division to divide [tex]\( x^5 - 105 \)[/tex] by [tex]\( x - 10 \)[/tex]. The remainder should be zero if [tex]\( x - 10 \)[/tex] is a factor of [tex]\( x^5 - 105 \)[/tex].

Let's perform polynomial long division:

         ____________________________

x - 10  |  x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 105

           - (x^5 - 10x^4)

           ____________________________

                       10x^4 + 0x^3 + 0x^2 + 0x - 105

                       - (10x^4 - 100x^3)

                       ____________________________

                                    100x^3 + 0x^2 + 0x - 105

                                    - (100x^3 - 1000x^2)

                                    ___________________________

                                                 1000x^2 + 0x - 105

                                                 - (1000x^2 - 10000x)

                                                 ___________________________

                                                              995x - 105

                                                              - (995x - 9950)

                                                              ___________________________

                                                                          9845

```

Since the remainder is a constant term, it's clear that [tex]\( x - 10 \)[/tex] is a factor of [tex]\( x^5 - 105 \)[/tex]. So, [tex]\( a = \frac{9845}{x - 10} \)[/tex].

Therefore, [tex]\( a = \frac{9845}{x - 10} \)[/tex].

d= dimes

q = quarters

d+q=26 coins

 rewrite as d=26-q

0.25q +0.10d=4.25

0.25q+0.10(26-q)=4.25

0.25q+2.6-0.10q=4.25

0.15q=1.65

q=11

d=26-11=15

11*0.25 = 2.75, 15*0.10=1.50, 2.75+1.50=4.25

 there are 11 quarters

Final answer:

The trapezoid forms a right-angled triangle with one additional rectangle. Its area is found to be 48m^2, and its approximate perimeter is 36.944m, by adding the lengths of all its sides together.

Explanation:

To find the perimeter and area of a trapezoid with two right angles and bases of 16m and 8m, we must first visualize the trapezoid. This trapezoid appears like a right-angled triangle with an additional rectangle attached to its hypotenuse.

We are given the hypotenuse of the altitude's right triangle is 4√5m, thanks to Pythagoras' theorem, we can find the two legs (which are the altitude h and the difference in bases). Let's call the altitude h and the difference in bases 'd'. Now, we know that the length of the longer leg of the right triangle is 16m - 8m = 8m.

Using the Pythagorean theorem where hypotenuse2 = altitude2 + difference in bases2, we have (4√5)2 = h2 + 82. Solving for 'h', we have h = √(80 - 64) = √16 = 4m. The area of a trapezoid is given by the formula A = (1/2) × (sum of the bases) × (height), which in this case is A = (1/2) × (16m + 8m) × 4m = 48m2.

For the perimeter, it can be calculated by adding the lengths of all sides. So, perimeter = 16m + 8m + 4m + 4√5m = 28m + 4√5m. To find the approximate value of 4√5m, we can calculate 4 × 2.236 (since √5 = 2.236), which gives us approximately 8.944m. Adding this to 28m gives us a perimeter of approximately 36.944m.

The correct answer is: B. 10111

To convert the number [tex]\(212_{\text{three}}\)[/tex] to base-two (binary) form, we need to first convert it to base-ten and then convert the base-ten number to base-two.

[tex]\(212_{\text{three}}\)[/tex] in base-ten:

[tex]\[ 212_{\text{three}} = 2 \cdot 3^2 + 1 \cdot 3^1 + 2 \cdot 3^0 = 18 + 3 + 2 = 23_{10} \][/tex]

Now, we convert [tex]\(23_{10}\)[/tex] to base-two:

[tex]\[ 23_{10} = 16 + 4 + 2 + 1 = 2^4 + 2^2 + 2^1 + 2^0 = 10111_{2} \][/tex]

Final answer:

The number 212 in base three (212three) converts to 17 in decimal, which is represented as 10001 in binary notation (base-two).

Explanation:

The number 212 in base three (2123) needs to be converted into base-two form (binary). Converting from base three to base two requires understanding each digit's place value in base three and then converting that to binary. Starting from the rightmost digit to the left, we have:

So, we have:

2123 = (1 × 32) + (2 × 31) + (2 × 30)

2123 = (1 × 9) + (2 × 3) + (2 × 1)

2123 = 9 + 6 + 2 = 17 in decimal.

Now we need to find the binary representation of 17:

So, the binary equivalent is 1(24)0(23)0(22)0(21)1(20), which is 100012.

2.8 radians is the answer.

Answer:

The battery was charged at a rate of 2.2 % per minute.

It took 35 minutes to fully charge the battery

Step-by-step explanation:

Initial amount of charge = 23%

Charge after 30 minutes = 89%

Percentage of charge gained in 30 minutes = 89-23

                                                                         = 66 %

Charging speed of battery = [tex]\frac{percentage\hspace{3}of\hspace{3}charge\hspace{3}gained}{time\hspace{3}taken\hspace{3}in\hspace{3}minutes}[/tex]

= [tex]\frac{66}{30}[/tex]

= 2.2 % per minute

2.2% charge takes 1 minutes

1% charge takes [tex]\frac{1}{2.2}[/tex] minutes

Initially it was 23% charged so it needs 77% more charge to be fully charged.

So, 77% charge takes [tex]\frac{1}{2.2}*77[/tex] minutes    

                         =  [tex]\frac{100}{2.2}[/tex] minutes

                         = 35 minutes