Evaluate the following using suitable identities.

a) 972

b) 102 X 105

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The surface area of the box shown is 186 square inches (186 in²).

Surface area is the measure of the area on a two-dimensional surface. It is the sum of the areas of all the shapes that make up a two-dimensional object. It can be used to calculate the area of a sphere, a cylinder, a rectangular prism, and more.

The box shown is a three-dimensional rectangular prism. The surface area (S.A.) of a rectangular prism is calculated by adding the area of each of its six faces. The area of each face is calculated by multiplying the length of the face by its width.

For the box shown, the length is 3 inches (3 in), the width is 12 inches (12 in), and the height is 10 inches (10 in). To find the surface area, we need to calculate the area of each face and add them together. The surface area of this box is calculated as follows:

Front and back faces: 3 in x 10 in = 30 in²

Left and right faces: 12 in x 10 in = 120 in²

Top and bottom faces: 3 in x 12 in = 36 in²

Total surface area: 30 in² + 120 in² + 36 in² = 186 in²

Therefore, the surface area of the box shown is 186 square inches (186 in²).

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Therefore, the solutions for θ in the interval 0° ≤ θ ≤ 360° are approximately 131.81° and 228.19°.

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, particularly right triangles. It is used to study and analyze various phenomena that involve periodic functions, such as waves, oscillations, and sound. Trigonometry also has practical applications in fields such as physics, engineering, navigation, and surveying. It involves the use of trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent to calculate the sides and angles of triangles and other geometric figures.

Here,

Let's solve for θ:

First, let's substitute u = cos(θ), so we have:

3u² - 5u - 4 = 0

Now we can use the quadratic formula:

u = [ -(-5) ± √((-5)² - 4(3)(-4))] / (2*3)

u = [ 5 ± √(49) ] / 6

u1 = (5 + 7) / 6 = 2

u2 = (5 - 7) / 6 = -2/3

Since the cosine function has a range of -1 ≤ cos(θ) ≤ 1, we can discard the solution u1 = 2.

Now we can solve for θ:

cos(θ) = u2 = -2/3

θ = cos⁻¹(-2/3) ≈ 131.81°  

θ = 360° - cos⁻¹(-2/3) ≈ 228.19°

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Width of the table = 3.5 feet , Length of the table = 5.5 feet and Perimeter of the table = 18 feet

What is Rectangle ?

A rectangle is a 2-dimensional shape with four straight sides, where opposite sides are parallel and equal in length. It has four right angles and its diagonals bisect each other. It is a type of quadrilateral, and its area is calculated by multiplying the length and width of the rectangle. It is commonly found in many geometric shapes and everyday objects such as doors, windows, and computer screens.

Let's convert all measurements to the same unit for easier calculation. Since the table width is given in inches, we can convert it to feet by dividing by 12 (since 1 foot = 12 inches).

Table width: 42 inches = 42÷12 = 3.5 feet

Let's denote the length of the table as 'l' feet. The perimeter of a rectangle is given by the formula:

Perimeter = 2 * (Length + Width)

Given that the perimeter is 18 feet, we can substitute the values into the formula:

18 = 2 * (l + 3.5)

Now, we can solve for 'l':

18 = 2l + 7 (distributing 2 to both terms in the parentheses)

2l = 18 - 7 (subtracting 7 from both sides)

2l = 11 (simplifying)

l = 11÷2 (dividing both sides by 2)

Therefore, Width of the table = 3.5 feet , Length of the table = 5.5 feet

and Perimeter of the table = 18 feet

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Option D : Based on the given data, the best prediction for the number of students who prefer ice cream at the college, which is approximately 1,440 students.

To make prediction of the number of students who prefer ice cream, we need to use the proportion of students who prefer ice cream from the sample to the entire population of 4,000 students.

Based on the given data, 81 out of 225 students prefer ice cream. To estimate the number of students who prefer ice cream in the entire college, we can use the ratio of students who prefer ice cream in the sample to the total number of students in the college.

The total number of students in the college is 4,000, so we can set up a proportion:

81/225 = x/4000

Solving for x, we get:

x = (81/225) * 4000 = 1,440

Therefore, the best prediction for the number of students who prefer ice cream in the entire college is 1,440.

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A college cafeteria is looking for a new dessert to offer its 4,000 students. The table shows the preference of 225 students.

Ice Cream Candy Cake Pie Cookies

81 9 72 36 27

Which statement is the best prediction about the scoops of ice cream the college will need?

A. The college will have about 480 students who prefer ice cream.

B. The college will have about 640 students who prefer ice cream.

C. The college will have about 1,280 students who prefer ice cream.

D. The college will have about 1,440 students who prefer ice cream.

Answer:

x-intercept(s):  

(

6

,

0

)

y-intercept(s):

(

0

,

−

9

)

Step-by-step explanation:

The expression will give 36 after evaluating.

What is an Expression?

In mathematics, an expression is a combination of numbers, variables, and mathematical operations that can be evaluated or simplified. An expression may contain one or more terms, which are separated by operators such as addition (+), subtraction (-), multiplication (*), division (/), and exponentiation (^).

Expressions can represent various mathematical concepts, such as equations, inequalities, functions, and polynomials.

Given:  m = -7 and n = 2,

We know that, the mode function converts the function into positive value.

So, |m - n| = |-7 - 2| = 9

Now,

4 |m - n| = 4 * 9 = 36

So the expression 4 |m - n| evaluates to 36 when m = -7 and n = 2.

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Answer:

2x+15

Step-by-step explanation:

Answer:

2x + 15

Step-by-step explanation:

First, we are doubling x.

We know that doubling is the same thing as multiplying by 2:

2x

Next, we are adding fifteen:

2x + 15

Answer:

Approxiamaltely 91.95 feet

Step-by-step explanation:

If the car travels 160 miles, it will take approximately 2.7 hours to cover that distance.

An inverse function is a function that "undoes" the effect of another function. More specifically, if we have a function f(x), its inverse function g(x) will undo the effect of f(x), such that if we apply f(x) and then g(x) to an input x, we get back the original input.

To find the inverse function of d(t) = 60t, we need to solve for t in terms of d. We start by setting d equal to the original function and solving for t:

d = 60t

Dividing both sides by 60, we get:

t = d/60

So the inverse function is t(d) = d/60.

To find t(160), we substitute 160 for d in the equation t(d) = d/60:

t(160) = 160/60

t(160) = 2.67 (rounded to 1 decimal place)

Therefore, if the car travels 160 miles, it will take approximately 2.7 hours to cover that distance.

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The number of smaller card that could be cut from the larger rectangular card is 64.

Yuri has a large rectangular card measuring 1.2 meters by 0.8 meters, he wants to cut it up to make small rectangular cards each measuring 13 cm by 11.5 cm.

Let's convert the units.

1.2 m = 120 cm

0.8 = 80 cm

Therefore, let's find the area of the rectangular card.

area of the large rectangular card = lw

where

area of the large rectangular card = 120 × 80

area of the large rectangular card = 9600 cm²

area of each small rectangular card = 11.5 × 13.5

area of each small rectangular card = 149.5 cm²

Therefore,

largest number of card that can be cut out from the larger card = 9600 / 149.5 = 64.2140468227

largest number of card that can be cut out from the larger card ≈ 64

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Using the pattern given, we found that the colour of the lightbulb in the 543rd position of a string of lightbulbs is green.

What is meant by the pattern?

A recurring arrangement of numbers, shapes, colours and other elements is known as a pattern. The Pattern can be connected to any kind of occasion or thing. When a group of numbers are arranged in a particular way, the arrangement is referred to as a pattern. Patterns can also occasionally be referred to as a series. The number of patterns can be limitless or finite. There are many distinct kinds of number patterns, including geometric, Fibonacci, and algebraic or arithmetic patterns. In mathematics, number patterns are quite prevalent.

Given,

The number of lights on a string of lights = 1000

The order in which the lights repeat is red, yellow, green and blue.

After every four lights, the order repeats.

We can get the pattern as follows.

The blue lights are in positions 4,8,12,........,4n.

We have to find the multiple of 4 close to 543.

Now we can check if 542 is a multiple of 4.

542/4 = 135.5

So it is not a multiple of 4.

Now check if 544 is a multiple of 4.

544/4 = 136

So the bulb in the 544th position is blue.

Then the bulb in the 543rd position should be green.

Therefore using the pattern given, we found that the colour of the lightbulb in the 543rd position of a string of lightbulbs is green.

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The area of the rectangle is 35.1 cm²

To find the area of a rectangle, you need to multiply its length by its width.

Given that the dimensions of the rectangle are 5.85 cm and 6 cm, respectively, the area of the rectangle is:

Area = Length x Width

Substitute the known values in the above equation, so, we have the following representation

Area = 5.85 cm x 6 cm

Evaluate the product

Area  = 35.1 cm²

Hence, the area of the rectangle to the nearest tenth is 35.1 cm²

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Complete question

The dimension of a rectangle is 5.85 cm by 6 cm. Find its area to the nearest tenth.

a) After one second in the air, the cannonball's height is:

[tex]h = -16(1)^2 + 128(1) + 6 = 118 feet[/tex]

b) 8 seconds pass while the cannonball is in the air (from when it is fired until it hits the ground).

The pace at which an object's position changes in relation to a frame of reference and time is what is meant by velocity. Although it may appear sophisticated, velocity is just the act of moving quickly in one direction. Since it is a vector quantity, the definition of velocity requires both magnitude (speed) and direction. Its SI equivalent is the meter per second (ms-1). A body is considered to be accelerating if its velocity changes, either in magnitude or direction.

from the question:

a) We can enter values of t into the equation[tex]h = -16t^2 + 128t + 6[/tex] and evaluate to obtain the height of the cannonball each second after it is shot.

The height of the cannonball at time t = 0 is as follows:

[tex]h = -16(0)^2 + 128(0) + 6 = 6 feet[/tex]

When t = 1, the cannonball has been in the air for 1 second, so its height is:

[tex]h = -16(1)^2 + 128(1) + 6 = 118 feet[/tex]

b) The graph of the function [tex]h = -16t^2 + 128t + 6[/tex] can be used to calculate how long the projectile has been in the air. When the cannonball is six feet above the ground, it is said to be in the air. The values of t must be determined when h is larger than or equal to 6.

Finding the values of t where the graph crosses the line y = 6 is one technique to do this. Another is to graph the function. Another method is to solve for t while setting h = 6:

[tex]-16t^2 + 128t + 6 = 6[/tex]

[tex]-16t^2 + 128t = 0[/tex]

-16t(t - 8) = 0

t = 0 or t = 8

Hence, the cannonball flies for 8 seconds (from when it is fired until it hits the ground).

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Answer:

Step-by-step explanation:

70/33.8=2.07100592

Answer:

b) The graph will shift down 5 units from its parent graph.

Step-by-step explanation:

Your answer is in the image.

The probability that the sum of the numbers when two dice are rolled is an odd number is given as follows:

0.5.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

Each dice has six possible outcomes, hence the total number of outcomes when two dice are rolled is given as follows:

6² = 36.

When we look at the pattern of the sum of two dice, from (1,1), (1,2), ..., to (6,6) we see that they alternate between even numbers and odd numbers, hence:

Hence the probability is given as follows:

p = 18/36 = 0.5.

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Answer:

10 cm

Step-by-step explanation:

Given:

A circular piece of paper of a radius of 20 cm is cut in half and each half is made into a hollow cone by joining the straight edges.

To find:

Find the slant height and base radius of each cone.

Solution:

The radius of the circular piece of paper = 20 cm

Finding the slant height of each cone:

Since the straight edges of the semi-circular part is joined together to form a cone

∴ Slant height of the cone so formed = Radius of the circular piece = 20 cm

Thus, the slant height of the cone is → 20 cm.

Finding the base radius of each cone:

Let "r" cm be the base radius of each cone.

We have,

[Length of the base of each cone] = [Length of each semi-circular part of the original circular piece of paper]

⇒

2

�

�

=

2

�

×

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

2

2πr=

2

2π × radius of original circle

⇒

2

�

=

2

×

20

2

2r=

2

2×20

⇒

2

�

=

20

2r=20

⇒

�

=

10

�

�

r=10cm

Thus, the base radius of each cone is → 10 cm.

Answer:

Step-by-step explanation:

1) To solve the equation √(6x) = x, we can first square both sides of the equation to eliminate the square root:

(√(6x))^2 = x^2

6x = x^2

Next, we can rearrange the equation into standard quadratic form by subtracting 6x from both sides:

x^2 - 6x = 0

Now, we can factor out an x from the left-hand side of the equation:

x(x - 6) = 0

Setting each factor equal to zero, we find two solutions:

x = 0 or x - 6 = 0

Therefore, the solutions to the equation √(6x) = x are x = 0 and x = 6.

2)To solve √(5x-6) = x, we can square both sides of the equation:

(√(5x-6))^2 = x^2

5x-6 = x^2

Rearranging this quadratic equation to the standard form ax^2 + bx + c = 0, we get:

x^2 - 5x + 6 = 0

This can be factored into:

(x - 2)(x - 3) = 0

Therefore, the solutions to the equation √(5x-6) = x are x = 2 and x = 3.

We should check if these solutions satisfy the original equation:

When x=2: √(5x-6) = √(5(2)-6) = √4 = 2, which satisfies the equation.

When x=3: √(5x-6) = √(5(3)-6) = √9 = 3, which also satisfies the equation.

Therefore, the solutions are x = 2 and x = 3.

3) To solve the equation √(2x-1) = x-3, we can square both sides of the equation to eliminate the square root:

(√(2x-1))^2 = (x-3)^2

Simplifying the left-hand side gives:

2x-1 = (x-3)^2

Expanding the right-hand side gives:

2x-1 = x^2 - 6x + 9

Rearranging the equation gives:

x^2 - 8x + 10 = 0

We can solve this quadratic equation :

x^2 - 8x + 10 = 0

Now we can solve this quadratic equation by using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -8, and c = 10. Substituting these values into the formula:

x = (-(-8) ± √((-8)^2 - 4(1)(10))) / 2(1)

Simplifying:

x = (8 ± √(64 - 40)) / 2

x = (8 ± √24) / 2

x = 4 ± √6

Therefore, the solutions to the equation √(2x - 1) = x - 3 are x = 4 + √6 and x = 4 - √6.

4) has no real solutions.

when you have to remove the square root you have to power the both sides of the equation.

1)6x= x^2

 0 = x^2 - 6x

0= x(x-6)

x=0 or x-6=0

            x=6

2) [tex]\sqrt{5x-6}[/tex] = x

   5x-6=x^2

   x^2 -5x+6=0

   (x-3)(x-2)=0

    x-3=0 or x-2= 0

    x=3 or x=2

3)

[tex]\sqrt{2x-1}[/tex] = x-3

2x-1 = (x-3)^2

2x-1= x^2 - 6x +9

x^2 -8x +10=0

this equation can't solve by factoring method easily. so we have to use the completing square method.

x^2-8x+10=0

x^2-8x= -10

x^2 - 8x+(-4)^2 = -10+ ( -4)^2

(x-4)^2 = 6

[tex]x= 4 +\sqrt{6}[/tex] or  [tex]x= 4 -\sqrt{6}[/tex]

4) [tex]x= 2 + \sqrt{2x-11}[/tex]

x^2= 4+4 [tex]\sqrt{2x-11}[/tex] + 2x-11

x^2= 7+ 2x +4[tex]\sqrt{2x-11}[/tex]

x^2-2x-7= 4[tex]\sqrt{2x-11}[/tex]

(x^2-2x-7)^2= 16 (2x-11)

x^4 +4x^2 +49-4x^3 + 28x - 14x^2 = 32x-176

x^4 - 4x^3 - 10x^2 - 4x + 225= 0

x(x^3 -4x^2 - 10x-4)+225=0

this is a power 4 th equation this equation can solve by normal steps.

The only pair of quadrilaterals that have congruent diagonals is a rectangle and a square, and the answer is C.

These attributes are:

C) A rectangle and a square: A rectangle's diagonals are congruent, and a square's diagonals are also congruent. The solution is C because a square is a particular case of a rectangle in which all sides are congruent.

Hence, a rectangle and a square are the only pair of quadrilaterals with congruent diagonals, and the answer is C.

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The length οf the diagοnal is 20 feet.

A rectangle is a type οf quadrilateral with parallel sides that are equal tο οne anοther and fοur vertices that are all 90 degrees apart. It is alsο knοwn as an equiangular quadrilateral fοr this reasοn. The term "parallelοgram" can alsο be used tο describe a rectangle because the οppοsing sides are equal and parallel.

Given that 16 feet and 12 feet make up a rectangle.

The diagοnal and twο cοnsecutive sides οf a rectangle make a right-angled triangle.

Tο find the diagοnal apply Pythagοrean theοrem.

The widely accepted geοmetric principle knοwn as the Pythagοrean Theοrem states that the square οn the hypοtenuse οf a right triangle equals the sum οf the squares οn its legs.

Draw a rectangle:

Cοnsider △ABC:

AB² + BC² = AC²

12² + 16² =AC²

AC² = 144+ 256

AC² = 400

Take square rοοt οn bοth sides:

AC = 20

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The measure of the angle A as required to be determined in the task content is; 16.7°.

It follows from the task content that the measure of the angle A is to be determined.

By using the trigonometric ratio;

tan (A) = BC / AC

Hence, by substitution; it follows that;

tan (A) = 6 / 20

Hence, by simplification; we have that;

tan A = 3/10

tan A = 0.3

Therefore, to determine A;

A = tan-¹ (0.3)

A = 16.7°.

Ultimately the measure of angle A as required to be determined is; 16.7°.

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Hence, in answering the stated question, we may say that As a result, the  function mass of the 30 cm long PVC pipe is roughly 733.48 grammes, rounded to the nearest tenth.

Mathematicians investigate numbers and their variants, equations and associated structures, forms and their locations, and prospective locations for these things. The term "function" refers to the relationship between a group of inputs, each with its own output. A function is a connection of inputs and outputs in which each input results in a single, distinct outcome. Each function has its own domain, codomain, or scope. The letter f is commonly used to denote functions (x). An x represents entry. On functions, one-to-one capabilities, so multiple capabilities, in capabilities, and on functions are the four basic types of accessible functions.

The volume of the PVC pipe can be estimated by subtracting the volumes of the outer and inner cylinders:

V = π/4 × (D² - d²) × L

where V denotes volume, is the mathematical constant pi (roughly equivalent to 3.14159), D denotes outer diameter, d denotes inner diameter, and L denotes pipe length.

V = π/4 × (6.5² - 5²) × 30\sV = 531.13 cm³

PVC pipe mass can be estimated by multiplying its volume by its density:

m = V × ρ

With the provided density, we get:

1.38 g/cm3 m = 733.48 g m = 531.13 cm3

As a result, the mass of the 30 cm long PVC pipe is roughly 733.48 grammes, rounded to the nearest tenth.

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A written history ​means the writing of history, such as the writing of history based on the critical examination of sources which is also known as historiography.

Also known as recorded history describes the historical events that have been recorded in a written form or other documented communication which are subsequently evaluated by historians using the historical method.

Historiography also includes the theory and history of historical writing. Modern historians strive to reconstruct a record of human activities and gain a deeper understanding of them.

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Answer:

Z = 2.91

Step-by-step explanation:

Mean:  123

Sample Size:  26

Standard Deviation:  7

x = 127

z = (x - μ) / (σ / [tex]\sqrt{n}[/tex] )

[tex]z=\frac{127-123}{\frac{7}{\sqrt{26} } } \\z= 2.91[/tex]

**If this is a z-table question, then P (x = 127) = P (z = 2.91) = 0.9982.**

Answer:

-10

Step-by-step explanation:

[tex]f(x) = 3x + 2[/tex]

Replace x in the function with the given value x = -4:

[tex]f( - 4) = 3 \times ( - 4) + 2 = - 12 + 2 = - 10[/tex]

Answer:

Step-by-step explanation:

A bandana is in the shape of a triangle. The base of the bandana is 30 in. wide and the height is 12 in.

Triangle Area = 1/2 b x h

1/2 30 x 12 =

15 x 12 = 180 in^2

Answer:

cm³ and ml

m³ and litre

This is the all you will need to know

Answer: -1280

Step-by-step explanation:

in a geometric sequence it either multiplies or divides negatively or positively, in this case -5 to -20 is multiplied 4 positively and so is -20 to -80

then -80x4= -320

-320x4= -1280

Answer:

$0.14

Step-by-step explanation:

[tex]8,308 - 7,998 = 310 \: miles[/tex]

We don't have information on how many gallons of gas Ms. Ward purchased, so we can't calculate the exact cost per mile. However, we can use the average fuel efficiency of her car to estimate the cost. Let's assume her car gets an average of 25 miles per gallon, which is a common fuel efficiency for a compact car.

To calculate how many gallons of gas she used, we can divide the number of miles driven by the fuel efficiency:

[tex]\frac{310 \: miles}{25 \: miles _{gallon} } = 12.4 \: gallons[/tex]

Assuming that the cost of gas was $3.50 per gallon, we can multiply the number of gallons used by the cost per gallon to find the total cost of gas:

[tex]12.4 \: gallons \times $3.50_{gallon} = $43.40[/tex]

To find the cost per mile, we can divide the total cost of gas by the number of miles driven:

[tex]\frac{$43.40}{310 \: miles} = $0.14_{mile} [/tex]

So Ms. Ward's cost per mile for this time period was approximately $0.14.

Answer:

Step-by-step explanation:

Miles travelled = 8,308 - 7,998 = 310 miles

Total cost = $26.79

Cost per mile[tex]=\frac{26.79}{310}[/tex] =0.0864

Ms. Ward's cost per mile is $0.09 (rounded to nearest cent).

Answer:

The area of the shaded region is 160.2 in² (to the nearest tenth).

Step-by-step explanation:

The area of the shaded region can be calculated by subtracting the area of the inner circle from the area of the outer circle.

[tex]\boxed{\begin{minipage}{4 cm}\underline{Area of a circle}\\\\$A=\pi r^2 $\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\\end{minipage}}[/tex]

If the circles have the same center, and the inner circle has a radius of 7 in, then the outer circle has a radius of:

[tex]\implies r = 7 + 3 = 10\; \sf in[/tex]

Therefore, the area of the shaded region is:

[tex]\begin{aligned}\sf Area_{shaded\;region}&=\sf Area_{outer\;circle}-Area_{inner\;circle}\\&=\pi \cdot 10^2 - \pi \cdot 7^2\\&=100\pi - 49 \pi\\&=51 \pi\\& = 160.221225...\\&=160.2\; \sf in^2\;(nearest\;tenth)\end{aligned}[/tex]