Using suitable identity evaluate the following:
(i) 98³
(ii) 188 × 212

If the rectangle is 9 cm longer than it's width , and the width is 1/7th of perimeter , then the length of rectangle is 15 cm and width of rectangle is  6 cm.

let the rectangle's length be  = l cm ;

and let the width of rectangle is  = w cm ;

the length is 9 cm longer than its width , that means ;

⇒ l = w + 9  ...equation(1)

and the width is 1/7 of perimeter , then

⇒ w = (1/7)(2l + 2w)

⇒ w = 2/7(l  + w)  .....equation(2) ;

substituting value of equation(1) in equation(2) ;

we get ;

⇒ w = (2/7) × ( w + 9 + w ) ;

on simplifying ,

we get ;

⇒ w = 6 .

from equation(1) ⇒ l = 6 + 9 = 15 .

Therefore , the length is 15 cm and width is 6 cm .

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following are true about ALL parallelograms are:

a. Opposite sides are congruent.

c. Opposite angles are congruent.

d. The diagonals bisect each other.

h. Consecutive angles are supplementary.

What is parallelogram?

A parallelogram is a straightforward quadrilateral with two sets of parallel sides in Euclidean geometry. A parallelogram's facing or opposing sides are of equal length, and its opposing angles are of similar size.

There are six crucial parallelogram characteristics to be aware of:

A rhombus is what we have when all of the sides of a parallelogram are congruent. Rhombi can use the characteristics of parallelograms.

We have a quadrilateral that is referred to as a trapezoid if just one pair of its sides are parallel. While the nonparallel sides are referred to as legs, the parallel sides are known as bases. We have what is referred to as an isosceles trapezoid if the legs are congruent.

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This is because a survey of healthy high school students who were chosen at random revealed that stress is more likely to produce muscle cramps.

(a) What kind of research is this?

"Observational study" is the study's type.

Reason: This is because a survey of healthy high school students who were chosen at random revealed that stress is more likely to produce muscle cramps.

Can this data be used to draw a causal conclusion about the link between elevated stress levels and muscle cramps?

No, an observational study was unable to draw any conclusions about the link between elevated stress levels and muscle cramps. The Chi-square test can be used to determine the relationship between increasing stress and cramps.

(c) Describe any potential confounding factors that could account for the correlation between greater stress and muscle cramps.

Muscle cramps may be impacted by the increased stress.

In this case, the factors under consideration are qualitative variables. Count the number of times that high school students have visited a doctor with a muscular ailment.

Confounding factors for stress may include anxiety over taking the exam, less sleep, spending more time studying, eating less, drinking more coffee, and other factors.

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The least amount of time spent per week on meetings for the sample of 25 CEOs is 12 hours.

The least amount of time spent per week on meetings for the sample of 25 CEOs can be found by looking at the lowest value in the data set. The data set provided above lists the time spent per week on meetings for a sample of 25 CEOs, and the least amount of time spent is 12 hours per week.

This is the smallest value in the data set, and it represents the minimum amount of time spent on meetings among the CEOs in the sample.

Note that this data is a sample of 25 CEOs and not the entire population of CEOs, so it is not representative of all CEOs. Additionally, the time spent per week on meetings is a quantitative variable, and it is continuous. Therefore, it is recommended to use measures of central tendency (mean or median) and measures of dispersion (standard deviation, range) to make inferences about the population.

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An inequality to represent the possible values for the height, h, for this triangle is 1ft≤h≤25ft

The area of a rectangle is expressed as;

Area = Length * height

A = Lh

If the area of a rectangle is at most 400 square centimeters, this is expressed as;

A ≤400

≤ means at most that the area of the rectangle cannot be greater than 400

Substitute the given value into the inequality expression

Lh ≤ 400

Given

L = 16

16h ≤ 400

Divide both sides by 16

16h/16 ≤ 400/16

h ≤ 400/16

h ≤ 25

Hence the possible values of the height are within the range 1ft≤h≤25ft since we cannot have a negative value for the dimension.

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Answer: Yes it's 15

Step-by-step explanation:

20/100 x 75 = 15

The area of the octagon is 0.875 units square.

Two unit squares share the same center. The overlapping region of the two squares is an octagon with perimeter 3.5.

Area of a triangle:

We need to know how to calculate the area of a triangle, in order to solve this question. The area of a triangle with base b and height h is [tex]\frac{hb}{2}[/tex]. An octagon is a 2D figure with 8 sides.

Let A, B, C, D, E, F, G, and H be eight consecutive vertices of the octagon. Let 0 be the common center of the square units.

Draw the segments OA, OB, OC, OD, OE, OF, OG and OH, connecting the vertices with the common center O of the squares.

The segments divide the octagon into 8 concurrent triangles.

In these triangles, AB, BC, CD, DE,EF, FG, GH, and HA are the bases; the height is 12 of a unit in all the triangles.

Thus, the area of the octagon=[tex]\frac{1}{2}[/tex]×[tex]\frac{1}{2}[/tex]×(AB+BC+CD+DE+EF+FG+GH+HA)

        =[tex]\frac{1}{4}[/tex]×(Perimeter of the hexagon)=14×3.5=3.54=0.875

Therefore, the area of the octagon is 0.875 units square.

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By  using the unit circle , we found that the value of Sec(0°) is (b) Sec(0°) = 1  .

The Unit Circle is defined as a circle that have a radius of 1 which is centered at the origin of a coordinate plane.

The angle in the unit circle are measured in radians.

The Secant function, denoted by Sec(x), is defined as 1/Cos(x)  ;

we have to find the value of Sec(0°) ;

So , at 0 degrees, the point (1,0) lies on the unit circle and

the x-coordinate of this point is 1 which is the cos(0) and

the value of Sec(0) = 1/cos(0) = 1/1 = 1

Therefore , the value of Sec(0°) is = 1  .

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The expression that would render the both the sides equivalent is 3x+2-2

In Algebra, the like terms are defined as the terms that contain the same variable which is raised to the same power. In algebraic like terms, only the numerical coefficients can vary. We can combine the like terms to simplify the algebraic expressions.

Given here: The expression 2+8x-16

To find its equivalent we have an incomplete expression 5(x-3)+1

Now 5(x-3)+1=5x-15+1

                     =5x-14

Thus Adding 3x+2-2 to this expression we get

5x-14+3x+2-2=2+8x-16

Therefore, we can deduce that the missing term must be 3x+2-2

Hence, The missing expression is 3x+2-2

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Answer: The value of 4.3x10^5 is 430000.

Step-by-step explanation:

To find the value of 4.3x10^5, you need to multiply 4.3 by 10 raised to the power of 5. This can be written as 4.310^5.

The value of 4.310^5 is 430000.

√131 lies between 11 and 12.

The numbers that include natural numbers and zero are called whole numbers.

Given is the square root of 131.

We can write the square root of 131 as -

√131 = 11.45

We can write the whole numbers as -

11 < 11.45 < 12

Therefore, √131 lies between 11 and 12.

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On applying the arithmetic operation of division, the expression 2.646 / 0.9 gives a result of 2.9.

What is arithmetic operation?

A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.

The expression given is -

2.646 / 0.9

The expression is given in decimal form.

Use the arithmetic operation of division.

First remove the decimal points.

Multiply the numerator and denominator by 1000 as there are three decimal points in the numerator -

(2.646 × 1000) / (0.9 × 1000)

Now divide the expression -

2646 / 900

= 2.94

Rounding off to one decimal point gives 2.9.

Therefore, the value is obtained as 2.9.

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The domain of g(x) - f(x) is all real numbers, or (-∞, ∞).

What is the domain?

The domain of a function is the set of all possible input values (x-values) for which the function is defined and produces a valid output (y-value). In other words, it is the set of values for which the function is valid and can be used without producing any errors or undefined results.

To perform the function operation g(x) - f(x), we need to subtract the function f(x) from the function g(x).

g(x) = x² - x + 1

f(x) = 4x + 3

g(x) - f(x) = x² - x + 1 - (4x + 3)

g(x) - f(x) = x² - 5x - 2

The domain of a function is the set of all possible input values (x-values) for which the function is defined and produces a valid output (y-value).

For the function g(x) - f(x) = x² - 5x - 2, there are no restrictions on the input values, meaning that any value of x can be plugged into the function to produce a valid output.

Therefore, the domain of g(x) - f(x) is all real numbers, or (-∞, ∞).

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The possible amounts of time for which they could rent the meeting room is 1.71 hours

The question states that

This implies that

35x + 5.80x ≤69.80

Solving the inequality to have

40.8x ≤69.80

Making x the subject of the relation we have

x≤1.71

In conclusion it takes the college 1.71 of time for them to rent the meeting room

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The area of the triangle formed by connecting the three points is 6 square units. The result is obtained by using the determinant method.

The area of a triangle in coordinate geometry can be found by using the determinant formula.

[tex]Area \: of \: triangle = \frac{1}{2} \left[\begin{array}{ccc}x_{1}&y_{1}&1\\x_{2}&y_{2}&1\\x_{3}&y_{3}&1\end{array}\right][/tex]

The three points of a triangle are:

Using the determinant formula, the area of the triangle is

[tex]= \frac{1}{2} \left[\begin{array}{ccc}2&2&1\\4&2&1\\3&3&1\end{array}\right][/tex]

= ½ |2(2-(3)) - 4(2-(3)) + 3(2-2)|

= ½ |2(-6) - 4(-6) + 0|

= ½ |-12 + 24|

= ½ |12|

= 6 square units

Hence, the area of the triangle is 6 square units.

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95 and 50 are the measures of those two angles.

The properties of the triangle are: The sum of all the angles of a triangle (of all types) is equal to 180°. The sum of the length of the two sides of a triangle is greater than the length of the third side. In the same way, the difference between the two sides of a triangle is less than the length of the third side.

given, One angle of a triangle measures 35°. The other two angles are in a ratio of 10:19.

Let second angle of the triangle is "x"

Thus, third angle of triangle will be = 10/19x

Since, sum of the three angles of triangle is 180 degrees

35 + x + 10/19x = 180

29x/19 = 145

x = 5 *19

x = 95

So, Second angle of triangle = x = 95

Third angle of triangle = 10/19x = 50

therefore, the measures of those two angles are 95 and 50.

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The equivalent of the expression -(-0.1p+4) is 0.1p-4

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value.

For example: 3(x + 2) and 3x + 6 are equivalent expressions because the value of both the expressions remains the same for any value of x. 3x + 6 = 3 × 4 + 6 = 18. and can also be written as 6(x² + 2y + 1) = 6x² + 12y + 6.

Similarly -( - 0.1p + 4) can be written in another for by multiplying the negative sign with every term in the parentheses.

therefore -1× -0.1p + -1×4 = 0.1p -4

This means -(-0.1p+4) and 0.1p -4 are equivalent

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For the function y = f (x),  the average rate of change on the interval - 1 ≤ x ≤ 3 is -2.

What is a function?

In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.

The graph of the function f(x) is graphed and the graph is given.

The average rate of change is a measure of how much the function changed per unit, on average, over that interval.

For a function f(x) at interval [a,b], the average rate of change is given as -

[f(b) - f(a)]/(b - a)

The interval values is given as - [-1,3]

The value for f(a) as seen in the graph is - 4.

The value for f(b) as seen in the graph is - -4.

Substitute the values in the equation -

Average rate of change = [f(b) - f(a)]/(b - a)

A = (-4 - 4)/[3 - (-1)]

A = -8/(3 + 1)

A = -8/4

A = -2

Therefore, the value for average rate of change is -2.

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Answer:

The correct answer is C f(2)

The sample size of the data which has a sample mean of 0.7 is 20.

What is mean?

Mean is a measurement of a probability distribution's central tendency along the median and mode. It also goes by the name "anticipated value."

Given:

Cathy has obtained a sample mean of 0. 7 and, but she cannot remember if the sample size was 20 or 25,

Calculate the sample size as shown below,

As we know that the variance would be positive,

s² ≥ 1 / (n - 1) [Σ xi²-nx²]

0 ≥ 1 / (n - 1) [Σ xi²-nx²]

[10 - n × x × 0.7²] ≥ 0

n × x × 0.7² ≤ 10

n ≤ 10 / (0.7)²

n ≤ 20.408

Thus, the sample size is 20.

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There are 2 values of k, 12 and -12, for which the polynomial x^2 + kx - 36 has two distinct integer roots.

The polynomial x^2 + kx - 36 has two distinct integer roots if and only if its discriminant is a perfect square.

The value of the discriminant is given by:

b^2 - 4ac = k^2 + 4 * 36 = k^2 + 144

So, for the polynomial x^2 + kx - 36 to have two distinct integer roots, k^2 + 144 must be a perfect square.

We know that the difference of two consecutive perfect squares is equal to an odd number, so we can start testing values of k by incrementing k by 1 and checking if k^2 + 144 is a perfect square.

Starting with k = 0, we have k^2 + 144 = 144, which is not a perfect square. For k = 1, k^2 + 144 = 145, which is also not a perfect square.

For k = 2, k^2 + 144 = 148, which is still not a perfect square.

We keep incrementing k and checking until we find a value of k such that k^2 + 144 is a perfect square.

When k = 12, we have k^2 + 144 = 168, which is a perfect square (12^2).

Therefore, the polynomial x^2 + 12x - 36 has two distinct integer roots.

We can repeat this process for negative values of k to see if the polynomial has two distinct integer roots for those values.

We find that for k = -12, the polynomial x^2 - 12x - 36 also has two distinct integer roots.

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The statement would e: A representative sample is one that accurately reflects a larger population.

The correct answer is an option (b)

We know that the sampling is used to find insights and observations about a population group.

A representative sample is nothing but a sample / a subset of a population that accurately reflect the characteristics of the larger group.

For example, consider a class of 50 students with 25 girls and 25 boys.

We can generate a representative sample that might include ten students: five girls and five boys.

As we know  the samples contain smaller, manageable versions of the larger group, they are useful in statistical analysis when population size is large.

Thus, a representative sample is used for obtaining insights and observations of the targeted population group.

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The complete question is:

A representative sample is one that accurately reflects a larger

a. dependent variable.

b. population.

c. control group.

d. correlation coefficient.

On multiplying 55 × 35, the answer is  1925.

The simplest way to start teaching multiplication is to establish the relationship between the concept and addition, which your pupils should already be familiar with. Make sure your students understand the first principle of multiplication—that it is just repeated addition—before continuing on.

Your youngster can learn that multiplication is actually just a means to repeatedly add multiple groups of the same number by using objects like blocks and small toys. For instance, ask your youngster to arrange the blocks into six groups of three by writing the issue 6 x 3 on a sheet of paper.

        5 5

    ×  3 5

 ------------

     2 7 5

 + 1 6 5 ×

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

    1925

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Answer: y,x = -2,3 is your answer. I hope this helps!

Step-by-step explanation:

The probability that a randomly selected battery of this kind lasts longer than 70 months is approximately 0.104(rounded off to the second decimal place).

The cumulative distribution function of an exponential distribution with a mean of 60 months is given by:

F(x) = 1 - e^(-x/60)

The probability that a randomly selected battery lasts longer than 70 months is given by:

P(X > 70) = 1 - F(70) = 1 - (1 - e^(-70/60)) = e^(-70/60) = 0.104

So, the probability that a randomly selected battery of this kind lasts longer than 70 months is approximately 0.104 (rounded off to the second decimal place).

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It takes 30 hours using the hose alone to fill the swimming pool.

Step 1: Make denominators the same

Step 2: Add or Subtract the numerators (keeping the denominator the same)

Step 3: Simplify the fraction

To add or subtract unlike fractions, the first step is to make denominators the same so that numerators can be added just like we do for like fractions.

Let [tex]x\\[/tex] represent the swimming pool be filled with hose alone.

Given that using the pipe alone it takes 12 hours. using both it takes 8 4/7 hours. According to given condition,

[tex]\frac{1}{x} + \frac{1}{12} = \frac{1}{8\frac{4}{7} } \\\frac{1}{x} = \frac{1}{\frac{60}{7} } - \frac{1}{12}[/tex]

[tex]\frac{1}{x} = \frac{7}{60} - \frac{1}{12}\\\\ \frac{1}{x} = \frac{7}{60} - \frac{5}{60}\\\\\frac{1}{x} = \frac{2}{60}\\\\ x = \frac{60}{2}\\\\\\ x= 30[/tex]

It takes 30 hours using the hose alone to fill the swimming pool.

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Answer:

To find the diagonal length, you can use the Pythagorean theorem.

If one side is 28 inches, and the board is a square, the other side must also be 28 inches.

a = side 1

b = side 2

c = diagonal length

a^2 + b^2 = c^2

(28)^2 + (28)^2 = c^2

784 + 784 = c^2

1568 = c^2

Square root both sides.

c = about 39.6 inches diagonal length