Andre came up with the following puzzle. “I am three years younger than my brother, and I am 2 years older than my sister. My mom’s age is one less than three times my brother’s age. When you add all our ages, you get 87. What are our ages?”



Jada writes this equation for the sum of the ages: (x)+(x+3)+(x−2)+3(x+3)−1=87 . Match each term of the equation with its meaning.

The required rate of change of the surface area of the cube is 66 sq,feet/second.

The momentum of a variable is represented by the rate of change, which is used to mathematically express the percentage change in value over a specified period of time. R = (D2 - D1)/T is a general formula for calculating the rate of change.

[tex]\frac{V}{dt}[/tex] = 1327 cubic feet per second

volume(V) = 528528 cubic feet

We know that

V = a^3

528528 = a^3

a = 80.85

Differentiate w, r, t

[tex]\frac{V}{dt}[/tex] = 3a^2[tex]\frac{da}{dt}[/tex]

1327 = 3(80.85)^2da/dt

da/dt = 0.068

Now,

Surface area(S) = 6a^2

Differentiate w, r, t

dS/dt = 12a(da/dt)

dS/dt = 12(80.85)(0.068)

dS/dt = 66 Sq.feet/second

Thus, required rate of change of the surface area of the cube is 66.

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The parallelogram's surface area is 24 square units as a result.

A quadrilateral with the opposing sides parallel is called a parallelogram (and therefore opposite angles equal). A parallelogram with all right angles is known as a rectangle, and a quadrilateral with equal sides is known as a rhombus. A quadrilateral is referred to as a parallelogram in geometry.

The opposite sides of a parallelogram are parallel and of equal length. Rhombus, rectangle, and square are a few instances of parallelograms.

Here points are a(−3, 0), b(−1, 5), c(7, 4), and d(5, −1).

= 1/2 [-3 (4 + 1) - 1 (- 7 - 20)]

= 1/2 [-3 (5) - 1 (-27)]

= 1/2 [-15 + 27]

= 1/2 [12]

= 6 sq units

Area of parallelogram abcd = Area of triangle abc + Area of triangle acd

= 18 + 6

= 24 sq units.

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Answer:that doesn’t seem possible to figure out

Step-by-step explanation:

?

the fox population in a certain region has an annual growth rate of 8 percent per year. it is estimated that the population in the year 2000 was 7600.

(a) function model for the t year after 2000 (exponential model):

[tex]p(t) = 7600 (1.08)^t[/tex]

The function is a mathematical phrase, rule, or law that establishes the connection between an independent variable and a dependent variable (the another variable).

A function is described as a relationship between a group of inputs and one output each. A function is, to put it simply, a connection between inputs in which each input is connected to exactly one output.

Population in 2000,

P(0) = 7600

Growth rate = 8% each year

(a) function model for the t year after 2000 (exponential model):

[tex]p(t) = 7600 (1.08)^t[/tex]

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Number Zero is neither positive nor negative.

A whole number, an integer can be positive, negative, or zero but cannot be a fraction or a decimal because it is a whole number. The natural numbers (1, 2, 3,...), their opposites (-1, -2, -3,...), and zero are all considered to be integers. Numerous branches of mathematics use integers, but number theory and algebra place a special emphasis on them.

Zero is not an integer that may be either positive or negative. It is regarded as a non-negative or neutral number. The absolute value of zero is also zero in terms of magnitude. |0| = 0. In the set of integers, zero is also regarded as the identity element of addition and the absorbing element of multiplication.

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The TQ is a chord, PR is a secant line and WU is a tangent line.

Circle is a two dimensional figure which consist of set of all the points which are at equal distance from a point which is fixed called the center of the circle.

A chord of a circle is the line segment joining two points on the circle.

Line segments TQ and QV are chords.

TV is also a chord, which is the diameter.

A secant line is a line which intersects the circle at two different points.

Line PR is a secant line which intersects the circle at T and Q.

A tangent line is a line which intersects the circle at exactly one point.

Line WU is a tangent line which intersects the circle at V.

Hence the chords in the given figure are TQ and QV, secant line is PR and tangent line is WU.

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The plot of 2 1/3, −5/6, and −3 1/2 on the number line is attached below

Generally, The uneven distribution of resources, opportunities, and advantages among different persons or groups in a society is what we mean when we talk about inequality.

This may manifest itself in a variety of ways, such as economic inequality (differences in income or wealth), social inequality (differences in status or access to resources), or political inequality (differences in power or influence) (differences in power or representation).

inequality, A statement in mathematics that establishes an order connection between two numbers or algebraic expressions, such as "greater than," "greater than or equal to," "less than," or "less than or equal to."

The mathematical symbol for "equals" has been substituted by an arrowhead in the statement "5x - 4 > 2x + 3," which gives it the appearance of an equation. It is an illustration of the concept of inequality. This indicates that the section on the left, which reads 5x minus 4, is more significant than the section on the right, which reads 2x plus 3.

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So the shortest distance he must walk to return to his house is approximately 10.8 miles.

The man walks 6 miles south and 9 miles east from his house. To find the shortest distance he must walk to return to his house, we need to use the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In this case, the man's walk can be represented by a right triangle, where the length of one side is 6 miles (the distance he walked south), the length of the other side is 9 miles (the distance he walked east), and the length of the hypotenuse is the shortest distance he must walk to return to his house.

Using the Pythagorean theorem, we can calculate the length of the hypotenuse as:

hypotenuse = √(6^2 + 9^2) = √(36 + 81) = √117

So the shortest distance he must walk to return to his house is approximately 10.8 miles.

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The total cost of the entertainment center after six years, if Zebina must pay it off, would be $9,203.19.

When a person purchases an item using a deferred payment plan with interest, the total cost of the item will be greater than the original purchase price due to the added interest.

To calculate the total cost of the entertainment center after six years, we need to use the formula for compound interest:

A = P(1 + r/n)^(nt)

where A is the total cost, P is the original purchase price, r is the annual interest rate (expressed as a decimal), n is the number of times the interest is compounded per year, and t is the number of years.

In this case, P = $2,798, r = 0.2195 (21.95% expressed as a decimal), n = 1 (because the interest is compounded annually), and t = 6. Plugging these values into the formula gives:

A = 2798(1 + 0.2195)^6

A = $9,203.19

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The particular solution of y = f(x) to the given differential equation [tex]\frac{dy}{dx} = y (xlnx)[/tex] with the initial condition f(1) = 4 is [tex]y = 4e^{\frac{1}{2}x^{2} lnx } - \frac{1}{4} x^{2} + \frac{1}{4}[/tex].

dy/dx = y(x lnx)

1/y dy = x lnx dx      Eqn (1)

Let u =  lnx, dv = x dx

Differentiate u = lnx

du = 1/x dx

Integrate dv = x dx

v = (1/x)x²

Now, integrate equation(1), we get

∫1/y dy = ∫x lnx dx

ln y = 1/2 x² ln x - 1/4 x² + C       Eqn (2)

Now, put the value of x and y to calculate the value of C

ln 4 = 1/2 ln 1 - 1/4 + C

ln 4 + 1/4 = C

Put the value of C in equation (2), and we get

ln y = 1/2 x² ln x - 1/4 x² + ln 4 + 1/4

[tex]y = 4e^{\frac{1}{2}x^{2} lnx } - \frac{1}{4} x^{2} + \frac{1}{4}[/tex]

--The given question is incorrect, the correct question is

''Find the particular solution y = f (x) to the given differential equation [tex]\frac{dy}{dx} = y (xlnx)[/tex] with the initial condition f(1) = 4."--

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As per the given frequency distribution, the value of variance is 9.8 and the value of standard deviation is 3.126

The term frequency distribution is defined as the distribution of a variable is the pattern of frequencies, meaning the set of all possible values and the frequencies associated with these values

Here we have the following frequency distribution with average number of pupils per teacher in some states of the United States.

While we looking into the table we have identified the value of variance as,

But in order to calculate the variance we must need the value of mean, it can be calculated as follows:

= >Mean (x) = ∑(f(x)) / ∑f

Apply the values on it, then we get

=> 660/42 = 15.71

Now, the value of variance is calculated as

=> Variance = ∑(f(x)²) / ∑f - (Mean²)

Apply the values on it, then we get,

=>  [10782/42] - [15.71] = 9.8

Then through the variance the value of standard deviation is calculated as,

=> standard deviation = √9.8 = 3.126

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Using half angle formula,

Half angle formulas are equations that contain the half angles of the trigonometric ratios.

Given that tanα = -1/2 and π/2 < α < π

To find sin(α/2), we use the half angle formula

sin(α/2) = √[(1 - cosα)/2]

Also, tan²α + 1 = sec²α = 1/cos²α

⇒ cosα = ±1/√(1 + tan²α)

Since π/2 < α < π

cosα = -1/√(1 + tan²α)

Substituing this into the equation for sin(α/2), we have

sin(α/2) = √[(1 - cosα)/2]

sin(α/2) = √[(1 - (-1/√(1 + tan²α)))/2]

Substituting  tanα = -1/2 into the equation, we have

sin(α/2) = √[(1 + 1/√(1 + tan²α))/2]

sin(α/2) = √[(1 + 1/√(1 + (-1/2)²))/2]

sin(α/2) = √[(1 + 1/√(1 + 1/4))/2]

sin(α/2) = √[(1 + 1/√(5/4))/2]

sin(α/2) = √[(1 + 1/√5/2)/2]

sin(α/2) = √[(1 + 2/√5)/2]

sin(α/2) = √[(√5 + 2)/√5)/2]

sin(α/2) = √[(√5 + 2)/2√5

sin(α/2) = √[(√5 + 2)/2√5 × √5/√5

sin(α/2) = √[5(√5 + 2)]/2 × 5

sin(α/2) = √[5(√5 + 2)]/10

So, sin(α/2) = √[5(√5 + 2)]/10

Using the half angle formula,  

tan(α/2) = sinα/(1 + cosα)

Using the trigonometric identity cosα = ±1/√(1 + tan²α)

cosα = ±1/√(1 + (-1/2)²)

cosα = ±1/√(1 + 1/4)

cosα = ±1/√(5/4)

cosα = ±1/√5/2

cosα = ±2/√5

Since π/2 < α < π

cosα = -2/√5

Also, since the trigonometric identity sinα = ±√(1 - cos²α)

So, sinα = ±√(1 - (-2/√5)²)

sinα = ±√(1 - 4/5)

sinα = ±1/√5

Since π/2 < α < π

sinα = +1/√5

Substituting cosα = -2/√5 and sinα = +1/√5 into the equation for tan(α/2), we have

tan(α/2) = sinα/(1 + cosα)

tan(α/2) = +1/√5/(1 + (-2/√5))

tan(α/2) = +1/√5/(1 - 2/√5)

tan(α/2) = +1/√5/(√5 - 2)/√5)

tan(α/2) = 1/(√5 - 2)

So, tan(α/2) = 1/(√5 - 2)

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The three comparisons are Beta had a higher percentage of listeners than group alpha, Beta had a higher mean number of hours listening than group alpha and Beta had a higher range of number of hours listening than group alpha.

The first comparison states that the Beta group had a higher percentage of listeners than the Alpha group. This means that out of all the people who listened to the radio station that day, a larger proportion of them belonged to the Beta group, compared to the Alpha group.

The second comparison is saying that the Beta group had a higher mean number of hours listening than the Alpha group. This means that on average, people in the Beta group listened to the radio station for more hours than people in the Alpha group.

The third comparison states that the Beta group had a higher range of number of hours listening than the Alpha group. This means that the difference between the minimum and maximum hours of listening in the Beta group is larger than the Alpha group. Which means that the listeners of Beta group have a greater variability in terms of hours of listening than the Alpha group.

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Answer:

y = 21/4 or 5[tex]\frac{1}{4}[/tex]

Step-by-step explanation:

step 1:  multiply -8 and 0.5y to get -4y

step 2:  multiply -8 and -3 to get 24

now you have distributed -8 and have removed parentheses to get:

-4y + 24 = 3

subtract 24 from each side to get:

-4y = -21

divide each side by -4 to get:

y = 21/4 or 5[tex]\frac{1}{4}[/tex]

Answer:

In this problem, we need to add a point to this graph which creates a right triangle with the given points.

The easiest way to do this is to find a point that is perpendicular to both of the points; more simply put, it has to have the x-coordinate of one of the given points and the y-coordinate of the other given point.

For example, adding the point (50, -14) to the graph satisfies this criteria, using the x-coordinate of (50, 25) and the y-coordinate of (130, -14). However, you could also use the y-coord of the first point and the x-coord of the second to add the point (130, 25). Both points in bold satisfy the criteria.

See the attached image for a graph of these points and the right triangles they create.

113875.44746 is principal and interest will he have already paid when his balloon payment is due.

What exactly does "interest rate" mean?

An interest rate informs you of how much borrowing will cost you and how much saving will pay off. The amount you pay for borrowing money, expressed as a percentage of the entire loan amount, is what you will pay in interest if you are a borrower.

                        The financial fee for borrowing money is called interest, and it is typically stated as a percentage, such as an annual percentage rate (APR). Lenders may charge interest for the use of their money, or borrowers may charge interest for the use of those funds.

=PMT(5.25%/12,12*30,-245500)*12*7

= 113875.44746

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The test statistic of -0.03 is compared to a critical value to determine if the null hypothesis can be rejected. If the test statistic is less than or equal to the critical value, the null hypothesis is rejected and the 90% figure is not accurate.

Calculate the sample proportion:

Sample Proportion

= 657/750

= 0.87

Calculate the test statistic:

Test Statistic

= 0.87 - 0.90

= -0.03

Compare the test statistic to a critical value:

If the test statistic is less than or equal to the critical value, then we can reject the null hypothesis. If the test statistic is greater than the critical value, then we cannot reject the null hypothesis.

The null hypothesis is that the percentage of adults in the region who drink milk is equal to 90%, while the alternative hypothesis is that the percentage is not equal to 90%. A sample of 750 adults is polled and the sample proportion is calculated to be 0.87.

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Answer:

To solve this problem, we will use the following steps:

Step 1: Convert the minutes to hours by dividing the minutes by 60: 45 minutes / 60 = 0.75 hours

Step 2: Add the hours and minutes together to find the total time: 3 hours + 0.75 hours = 3.75 hours

Step 3: Multiply the total time by the average speed to find the distance: 3.75 hours * 80 km/hour = 300 km

Final Answer: Tim drives 300 km.

The fundamental theorem of algebra states that you will have ‘n’ roots for an nth-degree polynomial, including multiplicity. So, your roots for f(x) = x^3 are 0 (multiplicity 3). The total number of roots is still 3 because you have to count 0 thrice.

A polynomial can be defined as an expression which consists of variables, constants and exponents, that are combined using mathematical operations such as addition, subtraction, multiplication and division. 
·[tex]f(x)= ax^{n}+bx^{n-1}+cx^{n-2} +k[/tex]

We can also phrase it as there are going to be ‘n’ values of x for which it will make polynomial f(x)=0.

For Example: [tex]p(x)= x^{2}-6x+9[/tex]

                   ⇒   n = 2

                   ∴ Total number of roots = 2

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The probability that her 10th roll is her last roll in the expression of a decimal to the nearest thousandth is 0.039.

As per information given, the condition of the event will be:

As we can find the value of each probability, the probability of the tenth roll is the last roll will be:

Last roll in tenth = 1 x (5/6)⁸ x 1/6

Last roll in tenth = 0.03876133989

Since, the question is asking for the nearest thousandth, the answer will be 0.039

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Answer:

  3√7 ≈ 7.937 units²

Step-by-step explanation:

You want to know the area of the isosceles triangle with congruent sides 4 units and base 6 units.

The perimeter of the triangle is ...

  P = 4 + 4 + 6 = 14

so the semiperimeter is ...

  s = P/2 = 14/2 = 7

By Heron's formula, the area of the triangle is ...

  A = √(s(s -a)(s -b)(s -c))

  A = √(7(7 -4)(7 -4)(7 -6)) = 3√7

The area of the triangle is 3√7 ≈ 7.937 square units.

__

Additional comment

You can also compute the area after finding the altitude (the length of the dashed line) using the Pythagorean theorem. The figure consists of two right triangles, each with leg 3 and hypotenuse 4. The missing leg (the triangle height) is then ...

  h = √(4² -3²) = √7

The area is ...

  A = 1/2bh = 1/2(6)(√7) = 3√7 . . . . square units

No, 22/7 is not a repeating decimal. The fraction 22/7 is an estimate for pi, but since pi is an irrational number, it does not repeat and 22/7 is not an exact representation of pi.

1. The ratio of the distance around a circle to the distance across a circle through its center is represented by the number pi.

2. The number pi is a decimal that does not repeat.

3. The fraction 22/7 is sometimes used as an estimate for pi.

4. 22/7 is not a repeating decimal because pi is an irrational number, meaning it does not repeat and 22/7 is not an exact representation of pi.

No, 22/7 is not a repeating decimal. The fraction 22/7 is an estimate for pi, but since pi is an irrational number, it does not repeat and 22/7 is not an exact representation of pi.

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Answer:

see the picture attached

Step-by-step explanation:

To graph this, we will count to a negative five on the graph. This means that we will move to the left five spaces and plot a point at (-5,0). Now, we will create a vertical line at -5 to encapsulate all of the values found at x = -5. This is essentially an infinite line, but the picture I included shows it on a 10 x 10 plane. Hope this helps!

Answer:

What is X= -5 graphed

Step-by-step explanation:

Graph x=-5 x = −5 x = - 5 Since x = −5 x = - 5 is a vertical line, there is no y-intercept and the slope is undefined.

Answer: Rotation around the origin

Step-by-step explanation:

If you rotate ABCDE around the point (0,0), you will see that it maps onto A'B'C'D'E'.

Angle A is 154.2857 degrees, angle B is 77.1429 degrees, and angle C is 25.7143 degrees.

What is the sum of the three angles in a triangle?

A triangle's angles add up to 180 degrees because one exterior angle is equal to the sum of the other two angles in the triangle.

Given that 2 angle A = 3 angle B = 6 angle C, we can set up the following equations:

2A = 3B

3B = 6C

We know that in any triangle, the sum of the three angles is always 180 degrees. We can set up the equation:

A + B + C = 180

We can substitute the values of A and B from the first two equations into this third equation:

2A + 3B + C = 180

2(3B) + 3B + C = 180

6C + C = 180

7C = 180

We can solve for C:

C = 180/7 = 25.7143

We can use the value of C to find the value of B:

B = 3C = 3(25.7143) = 77.1429

We can use the value of B to find the value of A:

A = 2B = 2(77.1429) = 154.2857

Therefore, angle A is 154.2857 degrees, angle B is 77.1429 degrees, and angle C is 25.7143 degrees.

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There are six different positive integers divisible by 4 that can be formed using each of the digits 1, 2, 3, and 4 at most once and no other digits: 12, 24, 32, 42, 43, and 124.

1. 12: 1 and 2

2. 24: 2 and 4

3. 32: 3 and 2

4. 42: 4 and 2

5. 43: 4 and 3

6. 124: 1, 2 and 4

The six different positive integers divisible by 4 that can be formed using each of the digits 1, 2, 3, and 4 at most once and no other digits are 12, 24, 32, 42, 43, and 124. To find these numbers, we begin by examining all the possibilities that can be formed using the digits 1, 2, 3, and 4. We can form 12, 24, 32, 42, 43 and 124 by combining different combinations of the digits. For example, 12 is made up of the digits 1 and 2. 24 is made up of 2 and 4. 32 is made up of 3 and 2. 42 is made up of 4 and 2.

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a. The value of h'(3) is 45.

b. The value of H'(1) is 36.

The complete table is in the attachment. h' and H are derivates from function h(x) and H(x). Using the Chain Rules to derivate the function.

Calculate the h'(x) function

h(x) = f(g(x))

h'(x) = d/dx {f(g(x))}

h'(x) = f' (g(x)) × g'(x)

h'(3) = f' (g(3)) × g'(3)

h'(3) = f'(2) × 9

h'(3) = 5 × 9

h'(3) = 45

Calculate the H'(x) function

H(x) = g(f(x))

H'(x) = d/dx {g(f(x))}

H'(x) = g' (f(x)) × f'(x)

H'(1) = g' (f(1)) × f'(1)

H'(1) = g'(3) × 4

H'(1) = 9 × 4

H'(1) = 36

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the height intercept would be 0 given the pool has no water at the beginning.

Which is the vertical intercept?

The vertical intercept (y-intercept) is found by evaluating the function when the input variable, x, is 0 and is always the same as the constant b. It can be thought of as the original value of the function. The horizontal intercept (x-intercept) is the value of the variable x when the function value is 0.

according to the constant dividend growth model

price = d1 / (r - g)

d1 = next dividend to be paid

r = cost of equity

g = growth rate

$2.5 / (0.1 - 0) = $25

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On solving the provide question, we can say that Area of the triangle, [tex]A = \sqrt{s( s - a )( s - b )( s - c )} \\[/tex] =  34.98 sq units

A triangle is a polygon since it has three sides and three vertices. It is one of the basic geometric shapes. The name given to a triangle containing the vertices A, B, and C is Triangle ABC. A unique plane and triangle in Euclidean geometry are discovered when the three points are not collinear. Three sides and three corners define a triangle as a polygon. The triangle's corners are defined as the locations where the three sides converge. 180 degrees is the result of multiplying three triangle angles.

here,

in this triangle

[tex]A = \sqrt{s( s - a )( s - b )( s - c )} \\[/tex]

and here

s = a+b+c/3

a = 10

b = 7

c =12

s = 10+7+12 / 3

s = 14.5

A = 34.98 sq units

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The equation x(2x+3) = x² + 1 is the quadratic equation and the remaining equation (x²)² + 1 = 2x³, (x +  1) + 8 = (x²)(x²), and (x²)³ = (x³)⁴ are not in quadratic equation.

The term quadratic equation in math is defined as an equation containing one term in which the unknown is squared and no term in which it is raised to a power higher than 2.

Here we have to find the equation that was not in the form of the quadratic equation.

As per the definition, we have know that the quadratic equation must be in the form of

=> ax² + bx + c = 0

So, here we have to look into the given expression, then w e have identified that only the equation x(2x+3) = x² + 1 is in the quadratic form.

Because the quadratic equation only have the highest power value of 2.

And the remaining are not the quadratic equation.

Therefore, option (a), (b) and (d) are the resulting options.

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