Thomas finished the 200-meter race in 25.7 seconds while Mikah finished in 28.1 seconds. How much faster did Thomas run the race than Mikah?

The graph of the function f(x) = 5(2)^x is an upward-sloping exponential curve that starts at (0, 5) and increases rapidly as x moves to the right, never crossing the x-axis.

The function f(x) = 5(2)^x represents exponential growth. Let's analyze its graph.

As x increases, the value of 2^x grows exponentially. Multiplying it by 5 further amplifies the growth. Here are a few key points to consider:

When x = 0, 2^0 = 1, so f(0) = 5(1) = 5. This is the y-intercept of the graph, meaning the function passes through the point (0, 5).

As x increases, 2^x grows rapidly. For positive values of x, the function will increase quickly. As x approaches positive infinity, 2^x grows without bound, resulting in the function also growing without bound.

For negative values of x, 2^x approaches zero. However, the function is multiplied by 5, so it will not reach zero. Instead, it will approach y = 0, but the graph will never touch or cross the x-axis.

The function is always positive since 2^x is positive for any value of x, and multiplying by 5 does not change the sign.

Based on these observations, we can conclude that the graph of f(x) = 5(2)^x will be an exponential growth curve that starts at (0, 5) and increases rapidly as x moves to the right, never crossing or touching the x-axis.

The graph will have a smooth curve that rises steeply as x increases. The rate of growth will be determined by the base, in this case, 2. The larger the base, the steeper the curve. The function will approach but never reach the x-axis as x approaches negative infinity.

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Comparing the √17 and 29/7 using <, > or = , we have √17 < 29/7.

Mathematical number comparison is the process or method of comparing two numbers to determine whether one is less, bigger, or equal to the other. The comparison symbols for numbers are "=", which stands for "equal to," "=", which stands for "greater than," and " ", which stands for "less than." Comparing amounts is a technique for figuring out how much to compare units in relation to a different standard or reference unit. A common reference point is necessary for two comparing units to be compared; otherwise, they cannot be compared.

Given,

Comparing the following using <, > or = ,

Numbers:

√17 and 29/7

For √17

Simplifying,

√17 = 4.123

For 29/7

29/7 = 4.14

√17 < 29/7

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Yes, there is an analogous ratio between 2:1 and 20:10.

We just cancel by a common factor. So 4:2=2:1 . The simplest representation of the ratio 4 to 2 is the ratio 2 to 1. Also, since each pair of numbers has the same relationship to one another, the ratios are equivalent.

By dividing the terms of each ratio by their greatest common factor, we may simplify both ratios to explain why.

As the greatest common factor for the ratio 2:1 is 1, additional simplification is not necessary.

The greatest common factor for the ratio 20:10 is 10. When we multiply both terms by 10, we get:

20 ÷ 10 : 10 ÷ 10

= 2 : 1

As a result, both ratios have the same reduced form, 2:1, making them equal.

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Answer:

look at the picture i have sent

To find the rate at which each plant grows, divide a change in height by the corresponding change in time.

Plant 1

From weeks 2 to 5, the plant grew from 8 inches to 20 inches.

The time interval is equal to 3 weeks, while the change in height is 12 inches. Then, the rate of growth is:

Plant 2

From weeks 2 to 6, the plant grew from 4 inches to 12 inches.

The time interval between weeks 2 and 6 is 4 weeks, while the change in height for that period was 8 inches. Then, the rate of growth is:

Since 4 is greater than 2, then the rate of growth of plant 1 is greater than the rate of growth of plant 2. Therefore, plant 1 grows faster.

===========================================================

Explanation:

To go from the first term to the second term, we add on some common difference d.

So,

B = A+d

B = (k+2)+d

4k-6 = k+2+d

4k-6-k-2 = d

d = 3k-8

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

Similarly, to go from the second term to the third term, we also add on d

C = B+d

C = (4k-6)+d

C = (4k-6)+(3k-8)

C = 7k-14

3k-2 = 7k-14

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

Let's solve for k

3k-2 = 7k-14

-2+14 = 7k-3k

12 = 4k

4k = 12

k = 12/4

k = 3  is the final answer

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

If k = 3, then we have these three terms:

The arithmetic progression (AP) is 5, 6, 7. The common difference is d = 1.

Note how d = 3k-8 = 3(3)-8 = 1

Hey There!! ~

The answer to this is: the upper bound for the length is \(21.5cm.\) Lower and Upper Bounds

The lower bound is the smallest value that will round up to the approximate value.

The upper bound is the smallest value that will round up to the next approximate value.

Ex:- a mass of 70 kg, rounded to the nearest 10 kg, The upper bound is 75 kg, because 75 kg is the smallest mass that would round up to 80kg.

Here , A length is measured as 21cm correct to 2 significant figures. We need to find what is the upper bound for the length . let's find out:

As discussed above , upper bound for any number will be the smallest value in decimals which will round up to next integer value . So , for 21 :

⇒  \(21.5cm.\)

21.5 cm on rounding off will give 22 cm . So , the upper bound for the length is \(21.5cm.\)

Hope It Helped!~

\(ItsNobody\)~

9. The curve passes through the point (-1, -3), which means

\(-3 = a(-1) + \dfrac b{-1} \implies a + b = 3\)

Compute the derivative.

\(y = ax + \dfrac bx \implies \dfrac{dy}{dx} = a - \dfrac b{x^2}\)

At the given point, the gradient is -7 so that

\(-7 = a - \dfrac b{(-1)^2} \implies a-b = -7\)

Eliminating \(b\), we find

\((a+b) + (a-b) = 3+(-7) \implies 2a = -4 \implies \boxed{a=-2}\)

Solve for \(b\).

\(a+b=3 \implies b=3-a \implies \boxed{b = 5}\)

10. Compute the derivative.

\(y = \dfrac{x^3}3 - \dfrac{5x^2}2 + 6x - 1 \implies \dfrac{dy}{dx} = x^2 - 5x + 6\)

Solve for \(x\) when the gradient is 2.

\(x^2 - 5x + 6 = 2\)

\(x^2 - 5x + 4 = 0\)

\((x - 1) (x - 4) = 0\)

\(\implies x=1 \text{ or } x=4\)

Evaluate \(y\) at each of these.

\(\boxed{x=1} \implies y = \dfrac{1^3}3 - \dfrac{5\cdot1^2}2 + 6\cdot1 - 1 = \boxed{y = \dfrac{17}6}\)

\(\boxed{x = 4} \implies y = \dfrac{4^3}3 - \dfrac{5\cdot4^2}2 + 6\cdot4 - 1 \implies \boxed{y = \dfrac{13}3}\)

11. a. Solve for \(x\) where both curves meet.

\(\dfrac{x^3}3 - 2x^2 - 8x + 5 = x + 5\)

\(\dfrac{x^3}3 - 2x^2 - 9x = 0\)

\(\dfrac x3 (x^2 - 6x - 27) = 0\)

\(\dfrac x3 (x - 9) (x + 3) = 0\)

\(\implies x = 0 \text{ or }x = 9 \text{ or } x = -3\)

Evaluate \(y\) at each of these.

\(A:~~~~ \boxed{x=0} \implies y=0+5 \implies \boxed{y=5}\)

\(B:~~~~ \boxed{x=9} \implies y=9+5 \implies \boxed{y=14}\)

\(C:~~~~ \boxed{x=-3} \implies y=-3+5 \implies \boxed{y=2}\)

11. b. Compute the derivative for the curve.

\(y = \dfrac{x^3}3 - 2x^2 - 8x + 5 \implies \dfrac{dy}{dx} = x^2 - 4x - 8\)

Evaluate the derivative at the \(x\)-coordinates of A, B, and C.

\(A: ~~~~ x=0 \implies \dfrac{dy}{dx} = 0^2-4\cdot0-8 \implies \boxed{\dfrac{dy}{dx} = -8}\)

\(B:~~~~ x=9 \implies \dfrac{dy}{dx} = 9^2-4\cdot9-8 \implies \boxed{\dfrac{dy}{dx} = 37}\)

\(C:~~~~ x=-3 \implies \dfrac{dy}{dx} = (-3)^2-4\cdot(-3)-8 \implies \boxed{\dfrac{dy}{dx} = 13}\)

12. a. Compute the derivative.

\(y = 4x^3 + 3x^2 - 6x - 1 \implies \boxed{\dfrac{dy}{dx} = 12x^2 + 6x - 6}\)

12. b. By completing the square, we have

\(12x^2 + 6x - 6 = 12 \left(x^2 + \dfrac x2\right) - 6 \\\\ ~~~~~~~~ = 12 \left(x^2 + \dfrac x2 + \dfrac1{4^2}\right) - 6 - \dfrac{12}{4^2} \\\\ ~~~~~~~~ = 12 \left(x + \dfrac14\right)^2 - \dfrac{27}4\)

so that

\(\dfrac{dy}{dx} = 12 \left(x + \dfrac14\right)^2 - \dfrac{27}4 \ge 0 \\\\ ~~~~ \implies 12 \left(x + \dfrac14\right)^2 \ge \dfrac{27}4 \\\\ ~~~~ \implies \left(x + \dfrac14\right)^2 \ge \dfrac{27}{48} = \dfrac9{16} \\\\ ~~~~ \implies \left|x + \dfrac14\right| \ge \sqrt{\dfrac9{16}} = \dfrac34 \\\\ ~~~~ \implies x+\dfrac14 \ge \dfrac34 \text{ or } -\left(x+\dfrac14\right) \ge \dfrac34 \\\\ ~~~~ \implies \boxed{x \ge \dfrac12 \text{ or } x \le -1}\)

13. a. Compute the derivative.

\(y = x^3 + x^2 - 16x - 16 \implies \boxed{\dfrac{dy}{dx} = 3x^2 - 2x - 16}\)

13. b. Complete the square.

\(3x^2 - 2x - 16 = 3 \left(x^2 - \dfrac{2x}3\right) - 16 \\\\ ~~~~~~~~ = 3 \left(x^2 - \dfrac{2x}3 + \dfrac1{3^2}\right) - 16 - \dfrac13 \\\\ ~~~~~~~~ = 3 \left(x - \dfrac13\right)^2 - \dfrac{49}3\)

Then

\(\dfrac{dy}{dx} = 3 \left(x - \dfrac13\right)^2 - \dfrac{49}3 \le 0 \\\\ ~~~~ \implies 3 \left(x - \dfrac13\right)^2 \le \dfrac{49}3 \\\\ ~~~~ \implies \left(x - \dfrac13\right)^2 \le \dfrac{49}9 \\\\ ~~~~ \implies \left|x - \dfrac13\right| \le \sqrt{\dfrac{49}9} = \dfrac73 \\\\ ~~~~ \implies x - \dfrac13 \le \dfrac73 \text{ or } -\left(x-\dfrac13\right) \le \dfrac73 \\\\ ~~~~ \implies \boxed{x \le 2 \text{ or } x \ge \dfrac83}\)

The area of the painted face with vertices at (-4, 2), (-4, -11), (4, 2), and (4, -11) which represents a rectangle is 104 square inches

A rectangle is a quadrilateral with the four interior angles that are each right angles, and opposite sides that are congruent and parallel.

The specified vertices of the face of the brick are;

(-4, 2), (-4, -11), (4, 2), and (4, -11)

The graph of the above point used to draw the quadrilateral representing the shape of the face in the question, created with MS Word, is attached

From the attached graph and drawing, we get;

Shape of the face of the doorstop = A rectangle

Length of the sides of the quadrilateral (rectangle)

Length = 2 - (-11) = 13

Width = 4 - (-4) = 8

Area of a rectangle  Length × Width

Area of the painted face of the brick, A = 13 × 8 = 104

The area of the rectangular painted face of the brick = 104 square inches

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Answer:

If two chords intersect in a circle, then the product of the segments of one chord equals the product of the segments of the other chord.

6(3x) = 4(4x + 1)

18x = 16x + 4

2x = 4

x = 2

The prove of the trig identity gives sin²θ + cos²θ = 1.

The proof of the trig identity is determined as follows;

Let's consider a right triangle with one acute angle θ.

Let the hypotenuse have length 1

Let the length of the adjacent side = cosθ

then, the opposite side length = sinθ

Apply Pythagorean theorem, to determine the hypotenuse side;

(sinθ)² + (cosθ)² = 1²

sin²θ + cos²θ = 1

Thus, we have proved the identity.

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Answer:

8 3/4

Step-by-step explanation:

First convert mixed number into improper fraction

3 3/4= multiply whole number by denominator, then add the numerator 3*4+3= 15/4

2 1/3= 2*3+1= 7/3

15/4*7/3= 8 3/4

Answer:

x = -5

Step-by-step explanation:

X -6= 11 - 11(x + 7)

Distribute

x-6 = 11 -11x -77

Combine like terms

x-6 = -11x -66

Add 11x to each side

x-6+11x = -11x-66+11x

12x-6 = -66

Add 6 to each side

12x -6+6 = -66+6

12x = -60

Divide by 12

12x/12 = -60/12

x = -5

It is not possible for a cross section of a cylinder to have a triangular shape

Cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces

A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes

A triangle is a shape formed when three straight lines meet

Cross section of the cylinder is depends on how we cut .The cross-section of a cylinder may be either circle, rectangle, or oval

Hence, the cross section of of a cylinder can not be a triangular shape

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Answer:

6

Step-by-step explanation:

y=mx+b

M=Slope

B=Yintercept (6 in this case)

Where the line crosses the Y axsis

y= -8x + 6

Hope this helps.

Answer:

Hi so when you write an equation for a line I like you start with finding the y-intercept (where the line crosses the y-axis) in this case it is 6, remember that. Now we need to find the slope. I like to just count over and down from the points they give us. In this case it is 8 down and over 1. 8/1 is just 8 and the line is pointing downwards so the line is negative.

Now we put it together: y=mx+b (m is slope and b is y-intercept)

y= -8x + 6

Hope this helped!

The flour and sugar coral that will be use to make the cookies is 2 1/4 cups.

A fraction is simply a piece of a whole. The number is represented mathematically as a quotient where the numerator and denominator are split. In a simple fraction, the numerator as well as the denominator are both integers.

In this situation, Tucker is making cookies the recipe called for 1/3/4 cups of flour and 1/2cup of sugar.

The flour and sugar coral that will be use to make the cookies will be:

= 1 3/4 + 1/2

= 2 1/4

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The surface area of a rectangular prism is 48 5/6 mi².

In Mathematics and Geometry, the surface area of a rectangular prism can be calculated and determined by using this mathematical equation or formula:

Surface area of a rectangular prism = 2(LH + LW + WH)

Where:

By substituting the given side lengths into the formula for the surface area of a rectangular prism, we have the following;

Surface area of rectangular prism = 2[6 × 2 1/3 + (1 1/4 × 6) + (1 1/4 × 2 1 /3)]

Surface area of rectangular prism = 2[14 + 15/2 + 35/12]

Surface area of rectangular prism = 48 5/6 mi².

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A = \(\frac{1}{2}h(b_{1} +b_{2} )\)

If A = 136 when \(b_{1}\) = 7 and h = 16, find \(b_{2}\)

We are given the equation that we are working with:

A = \(\frac{1}{2}h(b_{1} +b_{2} )\)

We are given certain values that the variables, in this case, are equal to:

A = 136

\(b_{1}\) = 7

h = 16

Substitute the given variables in the given equation for the corresponding numbers:

A = \(\frac{1}{2}h(b_{1} +b_{2} )\)

136 = \(\frac{1}{2}\) * 16 (7 + \(b_{2}\))

We know all the values in the equation except  \(b_{2}\). In order to find  \(b_{2}\) we must isolate it on one side of the equation

136 = 8 (7 + \(b_{2}\))

\(\frac{136}{8}\) = \(\frac{8 (7 + b_{2}) }{8}\)

17 = 7 + \(b_{2}\)

17 - 7 = 7 - 7 + \(b_{2}\)

10 =   \(b_{2}\)

If   \(b_{2}\) is equal to 10 then if we plug it back into the given equation both sides of the equation should equal each other. Remember to use PEMDAS

136 = \(\frac{1}{2}\) * 16 (7 + \(b_{2}\))

136 = \(\frac{1}{2}\) * 16 (7 + 10)

136 = \(\frac{1}{2}\) * 16 (17)

136 = 8 (17)

136 = 136

\(b_{2}\) = 10

The statement ' educators should ensure math instruction is only taught through structured activities rather than through everyday situations and routines is True. Option A

They include;

Thus, the statement ' educators should ensure math instruction is only taught through structured activities rather than through everyday situations and routines is True. Option A

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Answer:

Step-by-step explanation:

false

Answer:

y=-4/3x-9

=

−

4

3

x

−

9

Step-by-step explanation:

First we need to get the slope of the perpendicular line

The slope of a perpendicular line is equal to the negative inverse of the slope of the given line

y

=

m

x

+

b

y

=

3

4

x

−

2

⇒

m

=

3

4

m

'

=

−

1

m

⇒

m

'

=

−

1

3

4

⇒

m

'

=

−

4

3

Now that we have the slope, we need to find the y-intercept.

To find the y-intercept, we need to plug-in values of  

x

and  

y

that the line passes through

y

'

=

m

'

x

'

+

b

⇒

y

'

=

−

4

3

x

'

+

b

⇒

7

=

−

4

3

(

−

12

)

+

b

⇒

7

=

16

+

b

⇒

b

=

−

9

Therefore, the equation of the line is

y

=

−

4

3

x

−

9

Answer:

How 'bout them apples?

Step-by-step explanation:

In this data set, the only value that is an outlier is 100, since it is above the upper bound of 98.25.

To identify the outliers in the data set, we can use the concept of the interquartile range (IQR) and the 1.5×IQR criterion.

First, we need to find the first and third quartiles (Q1 and Q3) of the data set. To do this, we can order the data set from smallest to largest:

54, 72, 75, 81, 81, 82, 83, 83, 83, 85, 100

The median of the data set is the middle value, which is 82.

The lower half of the data set will consists of:

54, 72, 75, 81, 81

The median of the lower half is (72 + 75)/2 = 73.5, which is the value halfway between the two middle values.

The upper half of the data set will consists of:

83, 83, 83, 85, 100

The median of the upper half is (83 + 85)/2 = 84, which is the value halfway between the two middle values.

Therefore, the first quartile (Q1) is 73.5 and the third quartile (Q3) is 84.

The interquartile range (IQR) is the difference between Q3 and Q1:

IQR = Q3 - Q1 = 84 - 73.5 = 10.5

To identify the outliers in the data set using the 1.5×IQR criterion, we need to calculate the lower and upper bounds:

Lower bound = Q1 - 1.5×IQR = 73.5 - 1.5×10.5 = 57.75

Upper bound = Q3 + 1.5×IQR = 84 + 1.5×10.5 = 98.25

Any data point that is below the lower bound or above the upper bound is considered an outlier.

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Here we have a problem of rates of work, where we want to find the rate at which two people would work together, so we can find the time it takes to complete a given job when these two work together..

We will find that they need to work for 6/5 hours to do the laundry together.

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

We know that:

Then we can define:

Where the rates are computed as the quotient between 1 (1 total laundry completed) and the amount of time it takes to do it.

If they work together, the rate just adds up, so we will have:

rate = (M + R) = 1/3h + 1/2h

Now we need to write that as a single fraction, so we get:

rate =  1/3h + 1/2h = 2/6h + 3/6h = 5/6h

Now we can write an equation of the form:

rate*time = 1

Where the 1 again, means "one complete laundry done".

Then we have:

(5/6h)*T = 1

T = (6/5)h

This means that they need to work for (6/5) hours to do the laundry together.

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The true statements are:

(B) Both figures are congurent.

(C) The two figures have the same orientation but different positions.

(E) Corresponding angles and sides have the same measures.

In geometry, how an item is positioned in the space it occupies—such as a line, plane, or rigid body—is described in terms of its orientation, angular position, attitude, bearing, and direction.

It refers more particularly to the fictitious rotation required to shift an object from a reference placement to its present location.

To get to the current positioning, a rotation might not be sufficient.

It could be required to include a fictitious translation known as the object's location (or position, or linear position).

Together, the position and orientation completely explain where the object is situated in space.

Therefore, the true statements are:

(B) Both figures are congurent.

(C) The two figures have the same orientation but different positions.

(E) Corresponding angles and sides have the same measures.

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Answer:

Step-by-step explanation:

Factors:

#A

No

#B

Yes

#C

Yes

#D

No

Where variable r is the correlation coefficient.

1) In the scatter plot, when r is closer to 1 it means there is a strong evidence of positive association between the variables.

The scatter plot would be described as a strong positive relationship.

2) When r is near -1, in a scatter plot it indicates there is a strong evidence of negative association between the variables.

Therefore, when r is near -1, the scatter plot will look like a a downward slope, starting from the left.

3) In a scatter plot when r is close to 0, it indicates there is a weak or no relationship between the variables.

The graph when the r is near 0 the graph would be described as no linear relationship( there is weak evidence of relationship between the variables)

Answer:

Please type the question down

Step-by-step explanation:

Hard too see

Answer:

f inverse of x is equal to 3 to the power of the quantity x plus 6 end quantity minus 5 all over 4

Step-by-step explanation:

The function \(f(x)=\log_3 (4x+5)-6\) can be thought of as a series of steps, one operation at a time:

Start with x

Multiply by 4

Add 5

Take the \(\log_3\)

Subtract 6

That gives you a function value \(f(x)\).

To get the inverse function, read that list from the bottom up (in reverse order, using inverse operations at each step).

Start with x  (a bit confusing, because this  x  represents the function value you get at the end of the above list).

Add 6   (add is the inverse operation of subtract)

Raise 3 to the ...    \(3^{\text{result of previous step}\)

Subtract 5

Divide by 4

\(f^{-1}(x) = \frac{3^{x+6}-5}{4}\)

Let's test this out.  Find f(1).

\(f(1)=\log_3(4 \cdot 1 + 5)-6 =\log_3(9)-6=2-6=-4\)

Now put -4 into the inverse function.

\(f^{-1}(-4)=\frac{3^{-4+6}-5}{4}=\frac{3^2-5}{4}=\frac{9-5}{4}=\frac{4}{4} =1\)

The final result is the number we started with when we put 1 into f(x).

Finding an inverse is reversing the action of the function  f  by doing inverse operations in "backwards" order.

A lot of authors have you do this by switching  x  and  y  in the formula for a function, then solving for  x.

\(y = \log_3(4x+5)-6\text{ switch x and y}\\\\x = \log_3(4y+5)-6\\\\x+6 = \log_3(4y+5)\\\\3^{x+6}=4y+5\\\\3^{x+6}-5=4y\\\\\frac{3^{x+6}-5}{4}=y\)

Answer:

B-

Step-by-step explanation:

I took the test