1 mile divided by 2 laps = mi/lap

Answer:

Check below

Step-by-step explanation:

Laws of indices applied in this worksheet:

[tex]a^{n} . a^{m} = a^{n + m[/tex]

[tex]a^{n} .b^{n} = (ab)^{n}[/tex]

Example 1:  

 a)  [tex]x^{2} . x^{3} . x[/tex]    

                    = [tex]x^{2 + 3 + 1}[/tex]

                    = [tex]x^{6}[/tex]

b) [tex](-3.1)^{3}[/tex]×[tex](-3.1)^{4}[/tex]

                           = [tex](-3.1)^{3 + 4}[/tex]

                           = [tex](-3.1)^{7}[/tex]

You Try:

a) [tex]a^{5} . a^{2} . a^{2}[/tex]

                  = [tex]a^{5 + 2 + 2}[/tex]

                  = [tex]a^{9}[/tex]

b) [tex](-1)^{100} . (-1)^{37}[/tex]

                  = [tex](-1)^{100 + 37}[/tex]

                  = [tex](-1)^{137}[/tex]

Answer:

Step-by-step explanation:

One half ÷ 3 = 1/2 ÷ 3

To divide fractions, we need to multiply the first fraction by the reciprocal of the second fraction. So,

1/2 ÷ 3 = 1/2 x 1/3 = 1/6

Therefore, one half ÷ 3 is equal to one-sixth.

Based on the information provided, it can be concluded that by 10:00 p.m. the temperature was 1°.

First, let's list the information provided:

8:00 a.m. = -3

1:00 p.m. = +14

10:00 p.m. = - 10

Based on this, let's calculate the temperature by 10:00 p.m.

-3 + 14 = 11° by 1:00 p.m.

This means the temperature at 1:00 p.m. was 11° grades. Now, let's calculate the temperature at 10:00 p.m.

11 -10 = 1° by 10:00 p.m

This means that by 10:00 p.m. the temperature was 1°.

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The dimensions are Width = 10 inches and Length = 13 inches

Draw a rectangle.  If the length is 4 inches longer than three times its width, we can write the width as "w" and the length as the 3*width + 4

Area is (width)(Length) = (width)(3*width+4)

    W

 ----------

 |         |

 |         |

 |          |  L = 3w+4

 |         |

 |          |

 -----------

A = (3w+4)(w)

A = 3w² + 4w = 340.

(Problem states that area is 340 sq in)

Need to solve this quadratic equation

3y² + 4y - 340 = 0

Factor:

(w - 10) (w + 13) = 0

So

w - 10 = 0.    or.    w + 13 = 0

Solve these and get

w = 10. or. w = -13

Only one that makes sense in real life is the positive one.

So the dimensions are

Width = 10 inches

Length = 13 inches

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If jack has three packs of gum and sam has 4 packs of gum, then the total amount of gum they have together is 7 packs.

What is the addition?

The addition is putting numbers together to make a sum or total. The plus sign + tells you to add two numbers. = is an equals sign. It shows that things on either side of it are equal.

We have given,

Jack has 3 packs of gum and Sam has 4 packs of gum.

To find the total number of packs of gum they have together, we simply add their individual amounts:

3 packs (Jack) + 4 packs (Sam) = 7 packs

Therefore, the total amount of gum they have together is 7 packs.

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Answer:

4 hours

Step-by-step explanation:

By adding the x's on the graph, they practiced 2/4 + 3/4 + 3/4 + 1 + 1 hours, which is 4 hours

Answer:

(i) 90°

(ii) 10 cm

Step-by-step explanation:

What is a Rhombus?

A rhombus is a special parallelogram  that has all four sides equal. But unlike a square, the angles are not right angles

Properties of a rhombus

We are given AC = 16 cm, BD = 12 cm
Using property 3, we see that

AO = CO = AB/2 = 16/2 = 8 cm

BO = DO =12/2 = 6 cm

m∠AOB = 90°  Answer (i)

(ii) Since m∠AOB  = 90°, ΔAOB is a right triangle with AB as the hypotenuse and AO, BO as the legs of the right triangle

By the Pythagorean Theorem
AB² = AO² + BO²

AB² = 8² + 6²

AB² = 64 + 36

AB² = 100

AB = √100 = 10 cm   Answer (ii)

Answer:

Step-by-step explanation:

You have a rhombus ABCD with diagonals AC = 16 cm and BD = 12 cm. You want the magnitude of central angle AOB, and the length of one side, AB.

The diagonals of a rhombus bisect each other and cross at right angle. They divide the figure into four (4) congruent right triangles.

The central angle AOB has magnitude 90°.

Each of these right triangles will have leg lengths of 6 cm and 8 cm, which you recognize as having the ratio 3 : 4. These are the leg lengths of a 3:4:5 right triangle. In this rhombus, those right triangles have lengths 6 cm : 8 cm : 10 cm. The side AB of the rhombus is 10 cm.

__

Additional comment

If you don't remember the 3-4-5 right triangle, you can use the Pythagorean theorem to find the hypotenuse of the triangle with legs 6 cm and 8 cm:

  c² = a² +b²

  c² = 6² +8² = 36 +64 = 100

  c = √100 = 10

The side AB is 10 cm long.

Answer:

Step-by-step explanation:

The equation 12sec²x - 16 = 0 is not a polynomial equation and cannot be factored into the product of two or more polynomials.

To solve for x, we need to isolate sec²x by adding 16 to both sides and then dividing by 12:

12sec²x - 16 + 16 = 0 + 16

12sec²x = 16

sec²x = 16 / 12

sec²x = 4/3

Now that we have isolated sec²x, we can find x by finding the inverse secant of 4/3.

Note that the inverse secant is a multi-valued function, so there may be multiple solutions for x, which can be obtained using a calculator or tables of trigonometric functions.

False. The minimum of two independent exponential random variables is not necessarily exponential.

For two independent exponential random variables X and Y, with rate parameters λ1 and λ2 respectively, the minimum of X and Y is a random variable Z with the cumulative distribution function (CDF) given by

[tex]F_Z(z) = 1 - e^(-(λ1+λ2)z)[/tex].

The survival function of Z is given by

[tex]S_Z(z) = e^(-(λ1+λ2)z)[/tex].

Therefore, the probability density function of Z is

[tex]f_Z(z) = (λ1 + λ2)e^(-(λ1+λ2)z)[/tex].

This is not an exponential distribution because the form of the probability density function is not of the form [tex]f(z) = λe^(-λz)[/tex].

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The graph of G(x) is the graph of F(x) compressed vertically.

The equation of the graph of F(x) is given by x^2 and that of G(x) is given by 4/5 x^2.

The coefficient of x^2 in the equation of G(x) is 4/5, which is smaller than the coefficient of x^2 in the equation of F(x), which is 1.

This means that for a given value of x, the value of G(x) will always be less than the value of F(x). In other words, the graph of G(x) will be closer to the x-axis compared to the graph of F(x), which is equivalent to compressing the graph of F(x) vertically.

Therefore, the statement that best compares the graph of G(x) with the graph of F(x) is "The graph of G(x) is the graph of F(x) compressed vertically".

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Suppose that F(x) = x^2 and G(x) =4/5x^2 Which statement best compares the graph of G(x) with the graph of F(x)? The graph of G(x) is the graph of F(x) compressed vertically and flipped over the x-axis. The graph of G(x) is the graph of F(x) stretched vertically. The graph of G(x) is the graph of F(x) stretched vertically and flipped over the x-axis. The graph of G(x) is the graph of F(x) compressed vertically.

The process of simplifying square roots involves finding an equivalent expression for the square root that has smaller or more manageable numbers inside the root sign.

This can be achieved by using prime factorization, the properties of square roots, and rationalizing the denominator. The final result should be in the simplest form possible, without any radicals in the denominator, or any perfect square factors in the numerator.

A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 x 3 = 9. The symbol for square root is √. For example, √9 = 3.

The square root of a number can be either positive or negative, but for most purposes, the positive square root is used.

Additionally, there are also square roots of expressions, not just numbers, which can be simplified using similar methods

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The trigonometric ratios for each triangles are:

Please note that all angles B are the right angle, then:

cos B = 1; sin B = 0; tan B = -

The trigonometric ratio of sin, cos, tan, can be formulated as:

(please refer to the attached triangle below for reference)

cos A = side adjacent to A / Hypotenuse

Cos A = b/c

Sin A = side opposite to A / Hypotenuse

Sin A = a/b

Tan A = Side opposite to A / Side adjacent to A

Tan A = a/b

From these formulas, we can find that:

Triangle 1

Cos A = 40/50 = 4/5

Sin A = 30/50 = 3/5

Tan A = 30/40 = 3/4

Cos C = 30/50 = 3/5

Sin C = 40/50 = 4/5

Tan C = 40/30 = 4/3

Triangle 2

Cos A = 27/45 = 3/5

Sin A = 36/45 = 4/5

Tan A = 36/27 = 4/3

Cos C = 36/45 = 4/5

Sin C = 27/45 = 3/5

Tan C= 27/36 = 3/4

Triangle 3

Cos A = 15/17

Sin A = 8/15

Tan A = 8/15

Cos C = 8/17

Sin C = 15/17

Tan C = 15/8

Please note that all angle B on the three triangles are all a right angle, then:

Cos B = 1

Sin B = 0

Tan B = -

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If Arnav applies the Distributive Property to the expression 4 + 8(y + 3) + x, the expression that he will get is 4 + 8y + 24 + x or 8y + x + 28

As per the data given:

Arnav applies the Distributive Property to the expression 4 + 8(y + 3) + x.

Here we have to determine the equivalent expression after applying the Distributive Property to the expression 4 + 8(y + 3) + x.

Distributive property states that, for real numbers a, b, and c:

a (b + c) = ab + ac

An expression of the form A(B + C) can be solved as A × (B + C) = AB + AC.

From the given expression the part which we have to apply distributive property is 8(y + 3)

Here a = 8

b = y

c = 3

So, applying property in the expression

= 4 + 8(y + 3) + x

= 4 + 8y + 24 + x

= 8y + x + 28

Therefore, the equivalent expression is 4 + 8y + 24 + x or 8y + x + 28

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y= (1/9)x + b,  plug in the point to solve for b

5= (1/9)(-4) + b

b = 5 + 4/9 = 49/9

y= (1/9)x + 49/9

9y = x + 49

The equation of line in standard form and slope intercept form is,

⇒ y - 5 = - x

⇒ y = - x + 5

The equation of line in point-slope form passing through the points

(x₁ , y₁) and (x₂, y₂) with slope m is defined as;

⇒ y - y₁ = m (x - x₁)

Where, m = (y₂ - y₁) / (x₂ - x₁)

Given that;

Two points on the line are (0, 5) and (5, 0).

Now,

Since, The equation of line passes through the points (0, 5) and (5, 0).

So, We need to find the slope of the line.

Hence, Slope of the line is,

m = (y₂ - y₁) / (x₂ - x₁)

m = (0 - 5) / (5 - 0)

m = - 5 / 5

m = - 1

Thus, The equation of line with slope - 1 is,

⇒ y - 5 = - 1 (x - 0)

⇒ y - 5 = - x

⇒ y = - x + 5

Therefore, The equation of line passes through the points  (0, 5) and

(5, 0) will be;

⇒ y - 5 = - x

⇒ y = - x + 5

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Answer: The cubic parent function f(x)=x^3 has been transformed to form the function g(x)=-2(x+4)^3-5 in the following ways:

Vertical scaling: The function has been vertically scaled by a factor of -2, which means that the height of the function has been multiplied by -2.

Horizontal shift: The function has been horizontally shifted 4 units to the right. This is because the expression inside the parenthesis (x+4) can be seen as shifting the graph 4 units to the right.

Vertical shift: The function has been vertically shifted down by 5 units. This is because the constant term -5 has been added to the expression, which moves the graph down by 5 units.

So, the parent function f(x)=x^3 has been transformed by a vertical scaling by -2, followed by a horizontal shift 4 units to the right, and finally a vertical shift 5 units down to form the function g(x)=-2(x+4)^3-5.

Answer:

  x ≈ 4.19 meters

Step-by-step explanation:

You want the value of x given a (4x +2) by (2x +3) rectangle with a cutout corner that is (x+2) by (x-1) has an area of 194 square meters.

The difference in area between the enclosing rectangle and the missing corner is ...

  (4x +2)(2x +3) -(x +2)(x -1) = 194

  8x² +16x +6 -(x² +x -2) = 194

  7x² +15x +8 = 194

  7x² +15x -186 = 0 . . . . . . put in standard form

This is best solved using the quadratic formula:

  [tex]x=\dfrac{-b+\sqrt{b^2-4ac}}{2a}=\dfrac{-15+\sqrt{15^2-4(7)(-186)}}{2\cdot7}\\\\=\dfrac{-15+\sqrt{5433}}{14}\approx\boxed{4.193492}[/tex]

The value of x is about 4.19 meters.

Answer: -2.19 or 1.69

x = −2.18723​509003  

or

x = 1.687​23509003

Step-by-step explanation:

The area of the region bounded by the lines y = 3x - 6 and y = -2x + 8 between the x-axis is 12/5 units

We can find the intersection point between these two lines;

y = 3x - 6

y = -2x + 8

Set these two equations equal to each other.

3x - 6 = -2x + 8

5x - 6 = 8

Add 6 to both sides of the equation.

5x = 14

Divide both sides of the equation by 5.

x = 14/5  

Find the y-value where these points intersect by plugging this x-value back into either equation.

y = 3(14/5) - 6

Multiply and simplify.

y = 42/5 - 6

Multiply 6 by (5/5) to get common denominators.

y = 42/5 - 30/5  

Subtract and simplify.

y = 12/5

These two lines intersect at the point 12/5. This is the height of the triangle formed by these two lines and the x-axis.

Now let's find the roots of these equations (where they touch the x-axis) so we can determine the base of the triangle.

Set both equations equal to 0.

(I) 0 = 3x - 6  

Add 6 both sides of the equation.

6 = 3x

x = 2  

Set the second equation equal to 0.

(II) 0 = -2x + 8

Add 2x to both sides of the equation.

2x = 8

Divide both sides of the equation by 2.

x = 4

The base of the triangle is from (2,0) to (4,0), making it a length of 2 units.

The height of the triangle is 12/5 units.

Formula for the Area of a Triangle:

A = 1/2bh

Substitute 2 for b and 14/5 for h.

A = (1/2) · (2) · (12/5)

Multiply and simplify.

A = 12/5

The area of the region is 12/5 units

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Based on the information presented, the ratio represents 8th-grade baseball players to total athletes.

A ratio is a mathematical expression that shows the proportion or the number of a specific element in a category compared to the number of another element in the same category. For example, if you count the animals on a farm the ratio of cows to chickens can be 5:7, which means that for every 5 cows there are 7 chickens.

here, we have,

What does the ratio 5:73 mean?

5 : This is the number of baseball players who are in 8th-grade

73: This is the total of players. You get this number if you add all the numbers given.

Based on this, this ratio is comparing the number of baseball players and total players, which means the meaning of this ratio is " the ratio of 8th grade baseball players to total athletes".

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After solving the fraction answer is 4 7/12 but the Susi's answer is 5 7/1. So the Susi's answer is wrong.

The given expression is 8 1/3- 3 3/4

We first solve that expression.

We convert mixed fraction into improper fraction.

After conversion we equal the denominator of of both fraction.

Then we subtract the fraction.

After solving we check Susi's answer is right or wrong.

To convert 8 1/3 into improper fraction we multiply 8 and 3 then we add 1 in the answer of multiplication of 8 and 3.

8 1/3 = 25/3

To convert 3 3/4 into improper fraction we multiply 3 and 4 then we add 3 in the answer of multiplication of 3 and 4.

3 3/4 = 15/4

Now the expression is:

= 25/3 - 15/4

Multiply and divide by 12 on both side

= 25/3 × 12/12 - 15/4 × 12/12

= (25 × 4)/12 - (15 × 3)/12

= 100/12 - 45/12

= (100 - 45)/12

= 55/12

= 4 7/12

The simplified answer of given expression is 4 7/12 but the Susi's answer is 5 7/1. So the Susi's answer is wrong.

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If 6 times the 6th term of an A.P is equal to 9 times the 9th term, then 15th term of the A.P. is -2d with d as the common difference.

Let's call the common difference of the arithmetic progression (A.P.) "d". The nth term of an A.P. is given by the formula T = a + (n - 1)d, where a is the first term and n is the term number.

Using this formula, we can write two expressions for the 6th and 9th terms:

T₆ = a + (6 - 1)d = a + 5d

T₉ = a + (9 - 1)d = a + 8d

According to the problem, 6 times the 6th term of the A.P. is equal to 9 times the 9th term, so we can write the following equation:

6T₆ = 9T₉

Substituting the expressions for T₆ and T₉, we get:

6(a + 5d) = 9(a + 8d)

Expanding and solving for a, we get:

6a + 30d = 9a + 72d

Subtracting 6a from both sides:

24d = 3a + 72d

Subtracting 72d from both sides:

-48d = 3a

Dividing both sides by 3:

-16d = a

So, the first term of the A.P. is -16d.

To find the 15th term, we can use the formula for the nth term:

T₁₅ = a + (15 - 1)d = -16d + 14d = -2d

So, the 15th term of the A.P. is -2d.

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The graph of the first function is y = √(x - 5).

The graph of the second function is y =  |x - 1| + 4.

Suppose we have a function f(x).

f(x) ± d = Vertical upshift/downshift by d units (x, y ±d).

f(x ± c) = Horizontal left/right shift by c units (x - + c, y).

(a)f(x) = Vertical stretch for a > 0, vertical shrink a < 0. (x, ay).

f(bx) = Horizonatal compression b > 0, horizontal stretch for b < 0. (bx , y).

f(-x) = Reflection over y-axis, (-x, y).

-f(x) = Reflection over x-axis, (x, -y).

The graph of the first function is f(x) = y = √x.

Now, f(x - 5) would be the same function shifter 5 units to the right,

f(x) = √(x - 5).

The graph of the second function is  a modulus function shifted up 2 units and to right by 1 unit g(x) = |x - 1| + 2 and again g(x) + 2 =  |x - 1| + 4.

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Answer: 10

Step-by-step explanation:

ex: reciprocal of 2 would be 1/2 because 2 can be written as 2/1; you just switch the demoninator and numerator

Answer: (1,2.5), (2, 3.5), (3, 4.5)

Step-by-step explanation: The question states that the frame is one and a half inches wider than the picture, so the equation would be f = p+ 1.5. Add 1.5 to each value of p to find the value of f

The parallelogram EFGH has the following coordinates at point H: H(x, y) = (6, 4).

Parallelograms are quadrilaterals with two pairs of parallel sides and any quadrilateral can be generated by the knowledge of four vertices. Based on this fact, the following conditions must be observed:

[tex]\overline {EF} \parallel \overline {GH}[/tex]

The coordinates of the missing vertex can be found by the following vector sum:

F(x, y) - E(x, y) = H(x, y) - G(x, y)

H(x, y) = G(x, y) + F(x, y) - E(x, y)

H(x, y) = (4, 2) + (3, 5) - (1, 3)

H(x, y) = (6, 4)

The coordinates of point H in parallelogram EFGH are H(x, y) = (6, 4).

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The geometric mean of x and y is [tex]\sqrt{xy}[/tex]

So the geometric mean of 4 and 5 is [tex]\sqrt{4*5} = \sqrt{20} \approx 4.472[/tex]

Answer:

[tex]2\sqrt{5}[/tex]

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

Geometric mean of two numbers is the square root of their product.

The number we are looking for is:

Answer:

6(2x + 3y)

Step-by-step explanation:

             Two x plus three = 2x + 3y

      Six times of (2x + 3y) = 6 *(2x + 3y)