. A sheet of card is 20 centimetres by 30 centimetres. What is
the maximum number of circular name badges, of diameter
60 millimetres, that can be cut from this piece of card?

Answer:

  x = 6

Step-by-step explanation:

You want to know the value of x given a tangent to a circle from an external point has length x, while a secant from that point has length x-2 to the first intersection with the circle, and an additional 5 to the second intersection.

The product of the segments of the secant to the two points of intersection with the circle is equal to the square of the tangent segment.

  (x -2)(x -2 +5) = (x)² . . . . . use given values

  x² +x -6 = x² . . . . . . simplify

  x -6 = 0 . . . . . . subtract x²

  x = 6 . . . . . . add 6

__

Additional comment

There are rules for two secants, for a tangent and secant (as here), and for two chords internal to the circle. These can be easier to remember if you generalize them to a single rule: the product of the distance from the common point to the first circle intersection and the distance to the second circle intersection is the same for each segment.

For a tangent, the two circle intersections are the same point, hence the distance is squared. For chords, the two circle intersections are at the ends of the chords, and the common point is where the chords cross.

The final velocity of the train is 30m/s.

The rate at which an object's velocity with respect to time changes is referred to as acceleration. When an object's velocity changes, it is said to have or be accelerating.

The rate of change in velocity over time is another name for acceleration. The magnitude of an acceleration vector indicates how much the velocity will change, while the vector's direction indicates how the velocity will change—whether the speed is rising or falling, the vector's direction is changing, or some combination of the three.

Given, initial velocity = 0 m/s (as the train starts from rest)

          time = 60 seconds

Acceleration = v - u / t = 30 - 0/ 60 =0.5 m/s square

We know,

v = u + at

v = 0 + (0.5 x 60)v = 30 m/s

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The amount of sugar needed for the recipe to be tripled is 9 / 4 cups.

Caroline baked cookies from a recipe that called for 3/4 cups of sugar. she planned to triple the recipe. Therefore, the amount of sugar she needs is calculated as follows:

1 recipe = 3 / 4 cups of sugar

She wants to triple the recipe.

Hence,

1 recipe = 3 / 4 cups of sugar

3 recipe = ?

cross multiply

amount of sugar needed = 3 / 4 × 3

amount of sugar needed = 9 / 4

amount of sugar needed = 9 / 4 cups of sugars

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The Total Number of Hot Dogs sold were 240 and Cheese Burgers sold were 246

What is Linear Equation in Two Variables?

A linear equation in two variables is one that is stated in the form ax + by + c = 0, where a, b, and c are real integers and the coefficients of x and y, i.e. a and b, are not equal to zero.

Solution:

Let, Number of Hot Dogs Sold = x

Number of Cheese Burgers Sold = y

According to the Question:

x + y = 486 ----------(i)

2x + 3y = 1218 ----------(ii)

Multiplying Equation (i) by 2

2x + 2y = 972 ----------(iii)

Substracting Equation (iii) from Equation (ii)

y = 246

x = 240

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Answer:

Step-by-step explanation:

Point A = (5, 6)  is reflected across the x axis which is y =0.

When the point is reflected across the x axis, the x coordinate

remains positive, but the y coordinate becomes negative.

The 6 becomes negative (-6).

So the coordinates of A' are (5, -6) which is the first answer.

Answer:

8-(6+4)-12

8-12-6-4

Step-by-step explanation:

8-(6+4)-12 and 8-12-6-4 are equivalent expressions because they both compute to -6. This is because in both expressions, the 6+4 is calculated first, yielding 10. Then, the 8 is subtracted from 10, yielding -2. Finally, the 12 is subtracted from -2, yielding -6.

6+4)-8-12 is not an equivalent expression because it does not compute to -6. In this expression, the 6+4 is calculated first, yielding 10. Then, the 8 is subtracted from 10, yielding 2. Finally, the 12 is subtracted from 2, yielding -10.

8-6-12+4 is not an equivalent expression because it does not compute to -6. In this expression, the 8 is subtracted from 6, yielding -2. Then, the 12 is subtracted from -2, yielding -14. Finally, 4 is added to -14, yielding -10.

The expression for the height (h) in terms of radius r is h = 547/(∏r²).

What is a cylinder?

One of the most fundamental curvilinear geometric shapes, a cylinder has historically been a three-dimensional solid. It is regarded as a prism with a circle as its base in basic geometry. In several contemporary fields of geometry and topology, a cylinder can alternatively be characterized as an infinitely curved surface.

Given that the volume of the cylinder is 547 cubic inches.

The volume of a cylinder is V = ∏r²h

Where r is the radius of the cylinder.

h is the height of the cylinder.

Putting V = 547, r = r and h = h in V = ∏r²h

547 = ∏r²h

∏r²h = 547

Divide both sides by ∏r²:

h = 547/∏r².

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Thus ,after adding  3/4 and 7/10  we get 29/20 where 20 is common denominator

The smallest common multiple among the denominators of a group of fractions is known as the lowest common denominator (least common denominator) in mathematics. Fraction comparison, addition, and subtraction are made easier.

Here,

In order to add 3/4 and 7/10

we have to find LCM of 4 and 10

which comes out to be = 20

Thus,

addding =>  15+`14 /20

=>29/20

Thus ,after adding  3/4 and 7/10  we get 29/20 where 20 is common denominator

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Answer:

15

Step-by-step explanation:

When you see something like f (18), this means 18 needs to be plugged in for x

let's plug in 18

f (18) = [tex]\frac{2}{3} (18) + 3[/tex]

Let's solve

(18 × 2) ÷ 3 = 12

12 + 3 = 15

f (18) = 15

The dy/dx at point (2,8) is 4/3

From the information illustrated, x + 3y^(1/3) = y and point (2,8)

Find the derivative of this equation with respect to x and solve for dy/dx:

x + 3y^(1/3) = y

1 + [3 × 1/3 × y^(-2/3)]dy/dx = dy/dx

1 + [y^(-2/3)]dy/dx = dy/dx

dy/dx -  [y^(-2/3)]dy/dx = 1

dy/dx [1-  y^(-2/3)] = 1

dy/dx = 1/ [1 - y^(-2/3)]

At point (2,8), x = 2 and y = 8.

Substitute into dy/dx:

dy/dx = 1/ [1 - y^(-2/3)]

dy/dx = 1/ [1 - 8^(-2/3)]

dy/dx  = 1/ [1 - 1/4)]

dy/dx = 4/3

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To determine the regression equation, we need to fit a line to the data that best describes the relationship between the dependent variable and the independent variable. The equation of the line is typically expressed in the form:

where y is the dependent variable, x is the independent variable, a is the y-intercept (the point at which the line crosses the y-axis), and b is the slope of the line (the rate of change of y with respect to x).

If we determine that a = 10 and b = 0.5, the regression equation would be:

This equation describes the relationship between the dependent variable (days-on-the-market) and the independent variable (price). It tells us that for every unit increase in price, the number of days on the market is expected to increase by 0.5.

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The value of the differentiation dy / dx will be 13 / 5.

Calculus, also known as infinitesimal calculus or "the calculus of infinitesimals," is the mathematical study of continuous change, similar to how geometry studies shape and algebra studies generalizations of arithmetic operations.

Given that the expression is 5 - 2x²+y³ =-x³ -3y. The coordinate is (3,-2).

The value of dy / dx is calculated as,

5 - 2x²+y³ =-x³ -3y.

Differentiate the above equation with respect to x,

5 - 2x²+y³ =-x³ -3y

-4x + 3y² dy /dx = 3x² -3dy / dx

-4x - 3x² = (-3y² -3)dy/dx

dy /dx = [-4x - 3x²] / [ (-3y² -3) ]

Substitute the value of x and y in the above equation,

dy /dx = [-4x - 3x²] / [ (-3y² -3) ]

dy /dx = [-4 x 3- 3(3)²] / [ (-3(-2)² -3) ]

dy / dx = [ -12 - 27 ] / [ -15]

dy /dx = -39 /- 15 = 13 / 5

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Answer:

x=142

Step-by-step explanation:

47+95=x

x=142

the two angles equal x

Answer:

False.

Step-by-step explanation:

False. The study of the relationships among thermal energy, heat, and work is called thermodynamics, not dynamics. Dynamics is a branch of mechanics that deals with the motion of objects and the forces that cause that motion. Thermodynamics, on the other hand, is the study of the relationships between heat, work, and other forms of energy, and how these relationships can be used to understand the behavior of systems. It is a branch of physics that is concerned with the study of heat and its effects on matter.

The perimeter of the box has been obtained as 6a^3b^2 + 6ab^5.

We know that a box has the shape of a rectangle thus we would have to apply the formula for the area of a rectangle in this case and we would have;

P = 2(l + b)

P = perimeter

l =length

b = breadth

Then we have;

2[(-9a^3b^2 + 13ab^5) + (-10ab^5 + 12a^3b^2)]

Collect the like terms

2[(-9a^3b^2) + 12a^3b^2 + 13ab^5 +(-10ab^5)]

2[3a^3b^2 + 3 ab^5]

6a^3b^2 + 6ab^5

In  order of descending exponents of a, the perimeter is 6a^3b^2 + 6ab^5.

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The required number of bracelets that in the given question are 10.

In math, the meaning of a  equation is a numerical explanation that shows that two numerical articulations are equivalent. For example, 3x + 5 = 14 is a condition, where 3x + 5 and 14 are two articulations isolated by an 'equivalent' sign.

Carla divided 28 bracelets equally among 4 friends.

Number of bracelets each friends has = 28/4 = 7 bracelets

Then Carla gave 3 more bracelets to Magan.

So, Total bracelets Magan has = 7 + 3 = 10 bracelets

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Answer:

[tex]\ln (y) = -0.250 \ln (x)+0.223[/tex]

Step-by-step explanation:

Given table:

[tex]\begin{array}{|c|c|}\cline{1-2} \vphantom{\dfrac12}x & y\\\cline{1-2} \vphantom{\dfrac12} 1 & 1.250\\\cline{1-2} \vphantom{\dfrac12} 2 & 1.051\\\cline{1-2} \vphantom{\dfrac12} 3& 0.950\\\cline{1-2} \vphantom{\dfrac12} 4& 0.884\\\cline{1-2} \vphantom{\dfrac12} 5& 0.836\\\cline{1-2} \vphantom{\dfrac12} 6& 0.799\\\cline{1-2} \end{array}[/tex]

To convert  y = axⁿ  to linear form, take natural logs of both sides and rearrange:

[tex]\begin{aligned}y=ax^n \implies \ln y &= \ln ax^n\\\implies \ln y &= \ln a + \ln x^n\\\implies \ln y &= \ln a + n \ln x\\\implies \ln y &=n \ln x+ \ln a\end{aligned}[/tex]

This is in the straight-line form y = mx + c.

Substitute the first values of x and y from the table into the natural log formula and solve for ln(a):

[tex]\implies \ln 1.250= n \ln 1+\ln a[/tex]

[tex]\implies \ln 1.250= n (0)+\ln a[/tex]

[tex]\implies \ln a = \ln 1.250[/tex]

[tex]\implies \ln a = 0.223\; \sf (3 \; d.p.)[/tex]

Substitute the found value of ln(a) and the last values of x and y from the table into the  natural log formula and solve for n:

[tex]\implies \ln 0.799= n \ln 6+\ln 1.250[/tex]

[tex]\implies n \ln 6=\ln 0.799-\ln 1.250[/tex]

[tex]\implies n =\dfrac{\ln 0.799-\ln 1.250}{\ln 6}[/tex]

[tex]\implies n =-0.250\;\sf (3 \; d.p.)[/tex]

Substitute the found values of n and ln(a) into the formula to create an equation for ln(y):

[tex]\boxed{\ln (y) = -0.250 \ln (x)+0.223}[/tex]

It is assumed the statement is true for [tex]S_k[/tex]:

Show that [tex]S_{k+1}[/tex] is also divisible by 5:

The first part is divisible by 5 as we assumed in the beginning and the second part has a factor of 5 so the sum is divisible by 5.

Hence it makes [tex]S_{k+1}[/tex] true, so the proof is complete.

Answer:

The Pair Factors of 54 are (1, 54), (2, 27), (3, 18), and (6, 9) and its Prime Factors are 1, 2, 3, 6, 9, 18, 27, 54.

Step-by-step explanation:

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The Pair Factors of 54 are (1, 54), (2, 27), (3, 18), and (6, 9) and its Prime Factors are 1, 2, 3, 6, 9, 18, 27, 54.

Here, we have,

The factors of 54 are the numbers, that can divide 54 completely or evenly. When a pair of factors are multiplied together to produce the 54, then they are said to be pair factors. The factors divide the number completely. Hence, these factors cannot be a fraction.

Factors of 54: 1,2,3,6,9,18,27 and 54

Factor pairs of the number 54 are the whole numbers which are not a fraction or decimal number.

To find the factors of a number, 54, we will use the factorization method.

To find factors of 54, we have to divide 54 by all natural numbers from 1 to 54.

54 ÷ 1 = 54

54 ÷ 2 = 27

54 ÷ 3 = 18

54 ÷ 6 = 9

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Since ABCD has four equal angles and four equal sides, then it is a square.

The formula for the distance between two coordinates (x₁, y₁) and (x₂, y₂) is;  Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

We are given the coordinates;

A(0, 6)

B(6, 6)

C(6, 0)

D(0, 0)

Thus;

AB = √[(6 - 0)² + (6 - 6)²]

AB = 6

BC = √[(6 - 6)² + (0 - 6)²]

BC = 6

CD = √[(0 - 0)² + (0 - 6)²]

CD = 6

AD = √[(0 - 0)² + (0 - 6)²]

AD = 6

Thus, we conclude that all the sides are equal and it is a square.

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The shadow price of a constraint, the change in the objective function value for a unit change in (RHS) value of the constraint, with all other variables held constant.

A constraint is a condition of an optimization problem that should be satisfied the condition.

We can formulate this problem as a linear programming model (LPM):

Suppose X1 be the number of belt A produced per day, X2 be the number of belt B produced per day and X3 be the number of belt C produced per day

The objective function is to maximize the profit;

3X1 + 5X2 + 4X3.

The constraints are as :

The time spent on machine M1 must be less than or equal to 8 hours per day so we get;

2X1 + 3X2 <= 8

The time spent on machine M2 must be less than or equal to 10 hours per day so we get;

3X1 + 2X2 + 5X3 <= 10

The time spent on machine M3 must be less than or equal to 15 hours per day so we get

2X1 + 2X2 + 4X3 <= 15

Thus number of belts produced must be positive:

X1, X2, X3 >= 0

To solve this LPM using the simplex algorithm,

Also the objective function is to minimize the cost, and all constraints are in the form of "less than or equal to" a certain value.

The pivot row is the row that corresponds to the smallest positive ratio between (RHS) value and the corresponding pivot column element the constraint 2X1 + 3X2 ≤ 8.

Now Interpret the shadow prices.

The shadow price of a constraint is the change in the objective function value for a unit change in (RHS) value of the constraint, with all other variables held constant.

Here the value of the constraints in the LPM and how they contribute to the optimal solution.

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Answer:

0.2 g/mL

Step-by-step explanation:

The calculation for density is density=mass/volume.
With a mass of 10g and a volume of 50mL, you divide them to get 0.2

The triangles 1, 2, 4, 7 and 8 are right triangles by Pythagorean relationship.

Triangles are closed figures formed by three line segments, a right triangle is a triangle with one right angle. Right triangles have a long side known as hypotenuse (r) and two shorter sides known as legs (x, y), whose Pythagorean relationship is shown below:

r = √(x² + y²)

Now we proceed to check whether each triangle is a right triangle or not:

Case 1:

D = √(8² + 6²)

D = 10 (YES)

Case 2:

D = √(5² + 12²)

D = 13 (YES)

Case 3:

D = √(12² + 9²)

D = 15 (NO)

Case 4:

D = √(8² + 15²)

D = 17 (YES)

Case 5:

D = √(4² + 9²)

D ≈ 9.849 (NO)

Case 6:

D = √(14² + 5²)

D ≈ 14.866 (NO)

Case 7:

D = √(20² + 15²)

D = 25 (YES)

Case 8:

D = √(24² + 10²)

D = 26 (YES)

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Number of People who knows the Code are 6401

What is Linear Equation in Two Variable?

A linear equation in two variables is one that is stated in the form ax + by + c = 0, where a, b, and c are real integers and the coefficients of x and y, i.e. a and b, are not equal to zero.

Solution:

Let, Number of People who kows the Code be x

Number of Peoplee who do not know the Code be y

According to the Question:

x + y = 7338 ----------(i)

2x + 5y = 17487 ----------(ii)

Multiplying Equation (i) by 2

2x + 2y = 14676 ---------(iii)

Substracting Equation (iii) from Equation (ii)

3y = 2811

y = 937

This means x = 6401

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Mr. Muehlenweg S class used 1 3/4 cups baking soda for an experiment. If his students performed the experiment 6 1/2 times, he used 19/4 baking soda.

To subtract in mathematics is to take something away from a group or a number of objects. The group's total number of items decreases or becomes lower when we subtract from it. The components of a subtraction issue are the minuend, subtrahend, and difference.

Given,

Mr. Muehlenweg S class used 1 3/4 cups baking soda for an experiment.

Converting mixed fraction to improper function

1 3/4 = 7/4

If his students performed the experiment 6 1/2 times,

Converting mixed fraction to improper function

6 1/2 = 13/2

Subtracting,

7/4 - 13/2

7/4 - 26/4

19/4

Mr. Muehlenweg S class used 1 3/4 cups baking soda for an experiment. If his students performed the experiment 6 1/2 times, he used 19/4 baking soda.

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The equivalent expression of log(16 × 14) is log 16 + log 14.

Two expressions are said to be equivalent if they have the same value irrespective of the value of the variable(s) in them.

Equivalent expressions are expressions that have similar value or worth but do not look the same.

Therefore, the equivalent expression of log(16 × 14) can be found as follows:

using logarithm law,

The log of a product is the sum of the logs.

logₐ xy = logₐ x + logₐ y

Therefore,

log(16 × 14) = log 16 + log 14

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The correct stem-and-leaf plot for the data is:

i. Stem: 5,6,7,8,9 Leaf: 3, , 5 7 8 9, 1 2 4 9, 1

This plot shows the distribution of the data values, with the stems representing the tens place and the leaves representing the ones place. For example, the value 53 is represented by the stem 5 and the leaf 3, and the value 82 is represented by the stem 8 and the leaf 2.

The other three stem-and-leaf plots are incorrect because they do not accurately represent the distribution of the data.

A stem-and-leaf plot is a graphical representation of data that shows the distribution of values. The stems represent the tens place and the leaves represent the ones place. It is useful for identifying patterns and trends in the data.

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The trigonometric identities values are;

i. sin (A – B) = -0.15

ii. cos (A – B) = 0.5368

iii. tan (A + B) = -3.536

We are given;

cos A = 0.42

sin B = 0.73

Since cos A = 0.42 , then sin A = 0.58

Since sin B = 0.73, then cos B = 0.27

i) sin (A – B)

By trigonometric identities, we know that;

sin (A – B) = sinA cos B - cosA sinB

Thus;

sin (A – B) = (0.58 * 0.27) - (0.42 * 0.73)

sin (A – B) = -0.15

ii)  cos (A – B)

By trigonometric identities, we know that;

cos (A – B) = cosA cosB + sinA sinB

cos (A – B) = (0.42 * 0.27) + (0.58 * 0.73)

cos (A – B) = 0.5368

iii) tan (A + B)

By trigonometric identities, we know that;
(tan A + tan B)/(1 - tanAtanB)

tan A = sinA/cosA = 0.58/0.42 = 1.389

tan B = sinB/cosB = 0.73/0.58 = 1.259

Thus;

tan (A + B) = (1.389 + 1.259)/(1 - (1.389 * 1.259))

tan (A + B) = -3.536

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