Expand the fraction 72/125 .so that it has a denominator of 1000.

Proof by contradiction.

Let assume that when [tex]x[/tex] is an irrational number, then [tex]\dfrac{1}{x}[/tex] is a rational number.

If [tex]\dfrac{1}{x}[/tex] is a rational number, then it can be expressed as a fraction [tex]\dfrac{a}{b}[/tex] where [tex]a,b\in\mathbb{Z}[/tex].

[tex]\dfrac{1}{x}=\dfrac{a}{b} \Rightarrow x=\dfrac{b}{a}[/tex]

Since [tex]a,b\in\mathbb{Z}[/tex], the number [tex]x=\dfrac{b}{a}[/tex] is also a rational number. But this contardicts our initial assumption that [tex]x[/tex] is an irrational number. Therefore [tex]\dfrac{1}{x}[/tex] must be an irrational number.

The percentage of students who passed the class with a C or better is 82.1%. This is found by adding the number of students who earned A, B, or C grades, dividing by the total number of students, and multiplying by 100.

The subject is Mathematics, specifically a problem in percentages. First, we need to find the total number of students in the class. We add the number of students that earned each grade: 5 (A) + 8 (B) + 10 (C) + 3 (D) + 2 (F) = 28 students in total.  

Next, we need to identify the number of students who passed the class with a C or better. This includes the students who earned an A, B, or C. Adding these numbers together gets us 5 (A) + 8 (B) + 10 (C) = 23 students.  

Then, to find the percentage of students who passed the class, we divide the number of students who passed (23) by the total number of students (28) and then multiply by 100 to convert the result to a percentage. So the calculation is as follows: (23/28)*100 = 82.14. Rounded to one decimal place, that is 82.1%. Therefore, 82.1% of students passed the class with a grade of C or better.

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The percent of students who passed the course with a C or better is: 82.14%

To find the percent of students who passed the course with a C or better, we need to divide the number of students who passed by the total number of students and multiply by 100%.

The number of students who passed is 5 + 8 + 10 = 23.

The total number of students is 5 + 8 + 10 + 3 + 2 = 28.

Therefore, the percent of students who passed the course with a C or better is:

(23 / 28) * 100% = 82.14%

Rounding to one decimal place, the answer is 82.1%.

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To eliminate the parameter t in the equations x=t^2 and y=2+10t, solve [tex]t = \sqrt(x)[/tex] and substitute into the second equation to get [tex]y = 2 + 10\sqrt(x)[/tex]. This results in the cartesian equation [tex]y = 2 + 10\sqrt(x).[/tex]

To eliminate the parameter t and find the cartesian equation, follow these steps:

Start with the given parametric equations: x=t^2 and y=2+10t

Solve the first equation for t:

[tex]t = \sqrt(x)[/tex]

Substitute this expression for t into the second equation:

[tex]y = 2 + 10(\sqrt(x))[/tex]

Thus, the cartesian equation is [tex]y = 2 + 10 \sqrt(x).[/tex]

The algebraic rule for rotating a point [tex]\( 270^\circ \)[/tex] clockwise about the origin is:

[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]

Rotations in the coordinate plane can be performed using rotation matrices or by following specific rules based on the degree of rotation and the direction (clockwise or counterclockwise). For a rotation of [tex]\( 270^\circ \) or \( -90^\circ \) (since \( 270^\circ \)[/tex]  clockwise is equivalent to [tex]\( -90^\circ \)[/tex] counterclockwise), the rule can be derived as follows:

1.Starting Point:Consider a point [tex]\( (x, y) \)[/tex] in the coordinate plane.

2. 90° Rotation Counterclockwise (or 270° Clockwise):

  When you rotate a point [tex]\( 90^\circ \)[/tex] counterclockwise about the origin, the new position is [tex]\( (-y, x) \)[/tex].

3. 80° Rotation (second 90° Counterclockwise):**

  If you rotate [tex]\( (-y, x) \)[/tex] another [tex]\( 90^\circ \)[/tex] counterclockwise (making a total of 180°), the new position is [tex]\( (-x, -y) \)[/tex].

4.270° Rotation (third 90° Counterclockwise or 90° Clockwise):

Finally, rotating [tex]\( (-x, -y) \)[/tex] another [tex]\( 90^\circ \)[/tex] counterclockwise (or the original point [tex]\( (x, y) \) \( 90^\circ \)[/tex] clockwise), the new position is [tex]\( (y, -x) \).[/tex]

Therefore, the algebraic rule for rotating a point [tex]\( 270^\circ \)[/tex] clockwise about the origin is:

[tex]\[ (x, y) \rightarrow (y, -x) \][/tex]

This rule effectively swaps the x and y coordinates and negates the new x-coordinate, which corresponds to a [tex]\( 270^\circ \)[/tex] clockwise rotation.

Final answer:

The tens digit of a perfect square with a units digit of 6 could be any digit from 0 to 9. This is because when numbers ending in either 4 or 6 are squared, the tens digit can take any value in the range of 0-9 due to the cyclic nature of the squares.

Explanation:

The units digit of a perfect square can be 6, and this occurs when a number ending in either 4 or 6 is squared since 42 is 16 and 62 is 36. Considering this, the possible values for the tens digit of a perfect square with a units digit of 6 could be derived from these squares. For the tens digit, we must look at the squares of numbers ending in 4 or 6 and check the ten's place of the resulting product.

For example, if we consider 142 (which is 196) and 162 (which is 256), the tens digit can be either 9 or 5. Continuing with this pattern for numbers ending in 4 or 6 all the way to 942 and 962, we find that the tens digit is cyclic and can be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Therefore, all ten digits are possible for the tens place of a number whose square ends in 6.

Answer: y=(yx−16x)x+16

Step-by-step explanation:

Answer:

A quadratic equation is in the form of [tex]ax^2+bx+c =0[/tex]........[1], then the solution for this equation is given by:

[tex]x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex]

As per the statement:

The equation is given by:

[tex]h=-16t^2+25t+8[/tex]

where, h is the height in feet t seconds after it is thrown.

After the ball passes its maximum height, it comes down and then goes into the hoop at a height of 10 feet above the ground.

⇒h = 10 feet

then;

[tex]-16t^2+25t+8=10[/tex]

Subtract 10 from both sides we have;

[tex]16t^2-25t+2=0[/tex]

On comparing this equation with [1] we have;

a =16 , b =-25 and c =2

then;

[tex]t= \frac{25 \pm \sqrt{(-25)^2-4(16)(2)}}{2(16)}[/tex]

⇒[tex]t= \frac{25 \pm \sqrt{625-128}}{32}[/tex]

⇒[tex]t= \frac{25 \pm \sqrt{497}}{32}[/tex]

as we want the time when it was falling so ,

[tex]t= \frac{25 + \sqrt{497}}{32}[/tex]

Simplify:

[tex]t \approx 1.48[/tex] sec

Therefore, 1.48 sec long after it was thrown does it go into the hoop

Answer: Second option is correct.

Step-by-step explanation:

Since we have given that

61 ones times [tex]\frac{1}{10}[/tex]

which will be equal to

[tex]61\times \frac{1}{10}\\\\=6.1[/tex]

So, 6.1 will read as six point one

and

after decimal it is read as tenths.

So, it becomes,

61 tenths.

Hence, Second option is correct.

Answer: 304.

Step-by-step explanation:

The formula to calculate the sample standard deviation is given by :-

[tex]s=\sqrt{\dfrac{\sum(x-\overline{x})^2}{n-1}}[/tex]

, where x = sample element.

[tex]\overline{x}[/tex] = Sample mean

s=sample standard deviation.

n= Number of observations.

[tex]\sum(x-\overline{x})^2[/tex] = sum of the squared deviations from the sample mean

As per given , we have

s=4

n= 20

Substitute theses values in the above formula , we get

[tex]4=\sqrt{\dfrac{\sum(x-\overline{x})^2}{20-1}}[/tex]

[tex]4=\sqrt{\dfrac{\sum(x-\overline{x})^2}{19}}[/tex]

Square root on both sides , we get

[tex]\Righatrrow\ 16=\dfrac{\sum(x-\overline{x})^2}{19}\\\\\Righatrrow\ \sum(x-\overline{x})^2=16\times19=304[/tex]

Hence, the sum of the squared deviations from the sample mean is 304.

Sum of squared deviations: [tex]\( 16 \times (20 - 1) = 304 \),[/tex]  given a standard deviation of 4 and sample size of 20.

To find the sum of the squared deviations from the sample mean, we'll use the formula for variance. Since the standard deviation is the square root of the variance, we'll square the standard deviation to get the variance. Then, we'll multiply by the sample size minus 1 to get the sum of the squared deviations.

Given:

Standard deviation (σ) = 4

Sample size (n) = 20

First, let's find the variance:

[tex]\[ \text{Variance} = \sigma^2 = 4^2 = 16 \][/tex]

Now, we'll use the formula for the sum of squared deviations:

[tex]\[ \text{Sum of squared deviations} = \text{Variance} \times (n - 1) \]\[ \text{Sum of squared deviations} = 16 \times (20 - 1) \]\[ \text{Sum of squared deviations} = 16 \times 19 = 304 \][/tex]

So, the sum of the squared deviations from the sample mean is 304.