The Tortoise and the Hare is a fable about a race with the
moral, “Slow and steady wins the race.” The Tortoise and the
Hare decide to “race” across the United States from
Washington D.C. to Los Angeles.
a. Charles Darwin studied the tortoises when he was on
the Galapagos in 1835. He thought they moved relatively quickly. “One large one, I found by pacing, walked at the rate of 60 yards in 10 minutes” he wrote in Zoology Notes.
i How many inches per minute does the Galapagos tortoise walk?
ii. How long would it take a Galapagos tortoise to walk across the U.S.? Use the most
appropriate unit for time.
b. The Hare can run up to 30mph.
i. How many inches per minute does the Hare run?
ii. How long would it take the Hare to complete the race across the U.S.? Use the
most appropriate unit for time.
c. In the fable, the Hare, confident that he is so far head, relaxes and takes a nap. How long would the Hare need to nap for the Tortoise to pass him and win the race?

Part 1)  [tex]\displaystyle \frac{1}{n^2}[/tex] or 1/(n^2) goes in the box.

Part 2)  The series converges

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

Work Shown:

[tex]\displaystyle L = \lim_{n\to \infty} \frac{a_n}{b_n}\\\\\\\displaystyle L = \lim_{n\to \infty} \frac{\frac{n+3}{n^3-5n+8}}{\frac{1}{n^2}}\\\\\\\displaystyle L = \lim_{n\to \infty} \frac{n+3}{n^3-5n+8}\times\frac{n^2}{1}\\\\\\\displaystyle L = \lim_{n\to \infty} \frac{n^3+3n^2}{n^3-5n+8}\\\\\\[/tex]

[tex]\displaystyle L = \lim_{n\to \infty} \frac{\frac{n^3}{n^3}+\frac{3n^2}{n^3}}{\frac{n^3}{n^3}-\frac{5n}{n^3}+\frac{8}{n^3}}\\\\\\\displaystyle L = \lim_{n\to \infty} \frac{1+\frac{3}{n}}{1-\frac{5}{n^2}+\frac{8}{n^3}}\\\\\\\displaystyle L = \frac{1+0}{1-0+0}\\\\\\\displaystyle L = 1\\\\\\[/tex]

Because L is finite and positive, i.e. [tex]0 < L < \infty[/tex], this means that the original series given and the series  

[tex]\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2}[/tex]  

either converge together or diverge together due to the Limit Comparison Test.

But the simpler series is known to converge (p-series test).

Therefore, the original series converges as well.

The two series likely don't converge to the same value, but they both converge nonetheless.

Answer:

21

Step-by-step explanation:

(-4)(-2)+2(6+5)

(8)+2(11)

10+11

21

Hope this helps! :)

Answer:32

Step-by-step explanation: First, we do parentheses. 6+5 = 11. Next, we have to do multiplication (-4)(-2) is positive 8 because there are two negatives. 8+2 is 10 + 2(11) is 10+22 which is 32

Answer is b. ( -1, -5)

Answer is b. (-1, -5)

Step by step

Substitute the x and y values into both equations to find equality

 Answer b. Makes both equations equal

2x + y = -7

2(-1) + (-5) = -7

-2 -5 = -7

-7 = -7
it equals now let’s do the 2nd one

x - y = 4

-1 -(-5) = 4

4 = 4

This one equals too. I did the math on the other three answers and they did not equal. 

The regression equation based on the information given about the sugar contents will be 0.18x - 20.84.

It should be noted that from the information, the accompanying data are the caloric contents and the sugar contents of 11 high-fiber breakfast cereals.

The sugar content will be:

a. 160 calories

= 0.18x - 20.84.

= 0.18(160) - 20.84

= 7.96

b. 90 calories

= 0.18x - 20.84.

= 0.18(90) - 20.84

= -4.64

c. 175 calories

= 0.18x - 20.84.

= 0.18(175) - 20.84

= 10.66

d. 208 calories

= 0.18x - 20.84.

= 0.18(208) - 20.84

= 16.6

Therefore, the sugar contents is illustrated above.

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Since we know the sequence is quadratic, we expect the [tex]n[/tex]-th term to have the general form

[tex]x_n = an^2 + bn + c[/tex]

Plug in the first 3 known values of the sequence to form a system of equations.

[tex]x_1 = 3 = a + b + c[/tex]

[tex]x_2 = 12 = 4a + 2b + c[/tex]

[tex]x_3 = 25 = 9a + 3b + c[/tex]

Eliminating [tex]c[/tex] gives

[tex](4a + 2b + c) - (a + b + c) = 12 - 3 \implies 3a + b = 9[/tex]

[tex](9a + 3b + c) - (a + b + c) = 25 - 3 \implies 8a + 2b = 22 \implies 4a + b = 11[/tex]

Eliminating [tex]b[/tex] gives

[tex](4a + b) - (3a + b) = 11 - 9 \implies a = 2[/tex]

Solving for [tex]b,c[/tex], we get

[tex]4a + b = 11 \implies b = 11-4\cdot2 = 3[/tex]

[tex]a+b+c=3 \implies c=3-2-3 = -2[/tex]

So, the [tex]n[/tex]-th term of the sequence is

[tex]x_n = \boxed{2n^2 + 3n - 2}[/tex]

The graph is shown in the attached image.

Answer: 6.25

Step-by-step explanation: Here, you want to find how much of 6,298 pounds of food each person got. You can do this by dividing. 6,298/1,007 should get you the answer because it shows how the 6,298 pounds were distributed amongst the 1,007 people.

Answer:

a) x + 10 = 45

b) 3 = 6 - x

c) d = s * t

Step-by-step explanation: This is actually pretty simple. Let's put down one theorem.

SYMMETRIC THEOREM OF PROPERTIES: If a equals b, then b must equal a. Thus, a = b and b = a.

This is all we're really doing.

Take a) for example, 45 = x + 10. Let's label (45) as A, and (x + 10) as B. Thus we have A = B. Now we know by the Symmetric Property, we have B = A. Thus, substituting B and A again, we get x + 10 = 45.

Just keep doing this with all the equations.

Hope this helped!

Answer: 5 units

Step-by-step explanation:

x+2+2x-7=13

solve for x

3x=18

x=6

∴SR: 2x-7

=2(6)-7

=12-7

=5

Answer:

SR = 5

Step-by-step explanation:

[tex]TR=x+2+2x-7=3x-5[/tex]

[tex]3x-5=13[/tex]

[tex]3x=13+5[/tex]

[tex]x=18/3=6[/tex]

[tex]SR=2x-7=2(6)-7=12-7=5[/tex]

Hope this helps

Answer:

the slope of the line passing through the points (-4,-4) and (2, 5) m=3/2.

Step-by-step explanation:

Hello!

To find the slope between two points use the formula

The probability that they are female and use public transport is [tex]\mathbf{=\dfrac{1}{3}}[/tex]

The probability that they are female and walk/cycle to college is [tex]\mathbf{=\dfrac{1}{9}}[/tex]

Probability is an area of mathematics that deals with numerical representations of how probable an event is too probable to occur or how probable a statement is to be true.

Probability is synonymous with possibility. It is a mathematical field that is concerned with the occurrence of a random event. The value ranges from zero to one. The definition of probability is to predict the degree to which something is likely to occur.   To determine the likelihood of a particular event occurring, we must first determine the total number of possible outcomes.

Mathematically:

[tex]\mathbf{Probability = \dfrac{number \ of \ required \ outcomes}{total \ number \ of \ possible \ outcomes}}[/tex]

From the table in the image attached below.

The probability that they are female and use public transport is:

[tex]\mathbf{=\dfrac{9}{27}}[/tex]

[tex]\mathbf{=\dfrac{1}{3}}[/tex]

Also, the probability that they are female and walk/cycle to college is:

[tex]\mathbf{=\dfrac{3}{27}}[/tex]

[tex]\mathbf{=\dfrac{1}{9}}[/tex]

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Answer:

f(-6)= -19 and g(-4)= -187

Using the addition rule of the Sine function and the Cosine function, we obtain [tex]\cos\dfrac{x+y}{2}=\dfrac{b}{\sqrt{a^2+b^2}}[/tex].

Given that

[tex]\sin x + \sin y = a\\\cos x + \cos y = b[/tex]

Then using the above formulas, we get:

[tex]2\sin\frac{x+y}{2}\cos\frac{x-y}{2}=a[/tex]       (1)

[tex]2\cos\frac{x+y}{2}\cos\frac{x-y}{2}=b[/tex]       (2)

Dividing the equation (1) by (2), we get:

[tex]\dfrac{\sin\dfrac{x+y}{2}}{\cos\frac{x-y}{2}}=\dfrac{a}{b}\\\Longrightarrow \tan\dfrac{x+y}{2}=\dfrac{a}{b}[/tex]             (3)

Now, we know that  [tex]\cos\theta=\dfrac{1}{\sqrt{1+\tan^2\theta}}[/tex].

Thus, using the above formula, we get from (3):

[tex]\cos\dfrac{x+y}{2}=\dfrac{1}{\sqrt{1+\tan^2\dfrac{x+y}{2}}}\\\Longrightarrow \cos\dfrac{x+y}{2}=\dfrac{1}{\sqrt{1+\dfrac{a^2}{b^2}}}\\\Longrightarrow \cos\dfrac{x+y}{2}=\dfrac{b}{\sqrt{a^2+b^2}}[/tex]

Therefore, using the addition rule of the Sine function and the Cosine function, we obtain [tex]\cos\dfrac{x+y}{2}=\dfrac{b}{\sqrt{a^2+b^2}}[/tex].

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The length of the apothem of the regular pentagon shown is 5.2 meters

Represent the central angle of the regular pentagon using x

The value of the central angle of the regular pentagon is then calculated as:

x = 360/n

Where n represents the number of sides

i.e n = 5

So, we have:

x = 360/5

Evaluate the quotient

x = 72

Represents the apothem with y.

The apothem is then calculated as:

tan(x/2) = (Side length/2)/Apothem

This gives

tan(72/2) = (7.6/2)/y

Evaluate the quotient

tan(36) =3.8/y

Multiply both sides by y

y tan(36) = 3.8

Divide both sides by tan(36)

y = 3.8/tan(36)

Evaluate the quotient

y = 5.2

Hence, the length of the apothem of the regular pentagon shown is 5.2 meters

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Answer: $51156

Step-by-step explanation:

a = b x p%

how much will the price increase

a = 49,000 x 4.4%

= 49,000 x 0.044

= $2156

price of next model

49,000 + 2156 = $51156

Answer:

[tex]s=\frac{u}{r} -\frac{t}{r}[/tex]

Step-by-step explanation:

You have to move the variables (r and t) with u to isolate s.

[tex]rs+t=u[/tex]

   [tex]-t[/tex]       [tex]-t[/tex]

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

[tex]rs=u-t[/tex]

÷ [tex]r[/tex]       ÷ [tex]r[/tex]

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

[tex]s=\frac{u}{r} -\frac{t}{r}[/tex]

Considering it's discriminant, it is found that:

A. The classmate is wrong, as the discriminant is of zero, hence the equation has one solution.

B. The quadratic equation has 1 x-intercept.

A quadratic equation is modeled by:

y = ax^2 + bx + c

The discriminant is:

[tex]\Delta = b^2 - 4ac[/tex]

The solutions are as follows:

In this problem, the equation is:

y = 9x² - 6x + 1.

The coefficients are a = 9, b = -6 and c = 1, hence the discriminant is:

[tex]\Delta =(-6)^2 - 4(9)(1) = 36 - 36 = 0[/tex]

Since the discriminant is zero, the classmate is wrong, as it means that the equation has one solution = one x-intercept.

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Because the parabola intercepts the x-axis only once, we conclude that the discriminant is 0.

For a quadratic equation:

y = a*x^2 + b*x + c

The discriminant is:

D = b^2 - 4ac

In the graph we can see that the parabola intercepts the x-axis in its vertex, then the parabola has only one real zero, then we conclude that the discriminant is equal to zero.

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If the sample size is 300 and about 5% of the population has a particular genetic mutation then the standard deviation for the number of people with the genetic mutation in such groups of 300 is 3.77.

Given sample size be 300 and about 5% of the population has a particular genetic mutation.

We are required to find the standard deviation for the number of people with the genetic mutation in groups of 300.

Sample size=300

The standard deviation for the number of people with the genetic mutation is calculated as:

Standard deviation=[tex]\sqrt{np(1-p)}[/tex]

Use the known values in the above formula.

Standard deviation=[tex]\sqrt{300*0.05(1-0.05)}[/tex]

=[tex]\sqrt{14.25}[/tex]

=3.77

Hence if the sample size is 300 and about 5% of the population has a particular genetic mutation then the standard deviation for the number of people with the genetic mutation in such groups of 300 is 3.77.

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The required coordinates of the unit circle for s = -11π/6 is (√3/2, 1/2). Using trigonometric functions, the required answer is calculated.

There are six trigonometric functions.

They are sin θ, cos θ, tan θ, sec θ, cosec θ, and cot θ.

Where θ - 0 to 360° or 0 to 2π

For a unit circle, the coordinates are given in the form of trigonometric functions as below:

x = cos θ and y = sin θ

Here it is given that s = θ = -11π/6

So, on substituting,

x = cos θ = cos (-11π/6)

⇒ x = √3/2

y = sin θ = sin (-11π/6)

⇒ y = 1/2

Thus, the required coordinate pair is (√3/2, 1/2).

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Answer: 27°

Step-by-step explanation:

The easiest way to find the measure of angle Z is to know that in a rhombus, the diagonals are perpendicular bisectors of each other. This means that all the four angles in the center are 90°.

Since we know that all angles in a triangle add up to 180°, we can set the sum of z, 63, and 90 to be 180 and solve for z.

[tex]z+63+90=180\\z+153=180\\z=27[/tex]

Hence, z is 27°.

The equation of the function f(x) is f(x) = 2 sin(π/2(x + 6)) - 3

A sine function is represented as:

f(x) = A sin(B(x + C)) + D

Where

A = Amplitude

Period = 2π/B

C = Phase shift

D = Vertical shift

The requirements in the question are:

So, we can use the following assumptions

A = 2

Period = 4

C = 6

D = -3

So, we have:

f(x) = 2 sin(B(x + 6)) - 3

The value of B is

4 = 2π/B

This gives

B = π/2

So, we have:

f(x) = 2 sin(π/2(x + 6)) - 3

Using the representations in (a), we have:

See attachment for the graph of f(x)

Let θ = 3π

So, we have:

cos(3π)

This is calculated as:

cos(3π) = cos(2π + π)

Expand

cos(3π) = cos(2π) *cos(π) - sin(2π) *sin(π)

Evaluate

cos(3π) = -1

Let θ = 3π

So, we have:

sin(3π)

This is calculated as:

sin(3π) = sin(2π + π)

Expand

sin(3π) = sin(2π) *cos(π) + cos(2π) *sin(π)

Evaluate

sin(3π) = 0

Let θ = 3π

So, we have:

tan(2 * 3π)

tan(6π)

This is calculated as:

tan(6π) = tan(3π + 3π)

Evaluate

tan(3π) = 0

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Answer:

[tex]\displaystyle{x = \dfrac{\pi}{12}, \dfrac{\pi}{6}}[/tex]

Step-by-step explanation:

We are given the trigonometric equation of:

[tex]\displaystyle{\sin 4x = \dfrac{\sqrt{3}}{2}}[/tex]

Let u = 4x then:

[tex]\displaystyle{\sin u = \dfrac{\sqrt{3}}{2}}\\\\\displaystyle{\arcsin (\sin u) = \arcsin \left(\dfrac{\sqrt{3}}{2}\right)}\\\\\displaystyle{u= \arcsin \left(\dfrac{\sqrt{3}}{2}\right)}[/tex]

Find a measurement that makes sin(u) = √3/2 true within [0, π) which are u = 60° (π/3) and u = 120° (2π/3).

[tex]\displaystyle{u = \dfrac{\pi}{3}, \dfrac{2\pi}{3}}[/tex]

Convert u-term back to 4x:

[tex]\displaystyle{4x = \dfrac{\pi}{3}, \dfrac{2\pi}{3}}[/tex]

Divide both sides by 4:

[tex]\displaystyle{x = \dfrac{\pi}{12}, \dfrac{\pi}{6}}[/tex]

The interval is given to be 0 ≤ 4x < π therefore the new interval is 0 ≤ x < π/4 and these solutions are valid since they are still in the interval.

Therefore:

[tex]\displaystyle{x = \dfrac{\pi}{12}, \dfrac{\pi}{6}}[/tex]

Answer:

c = -75

Step-by-step explanation:

cd + (-6c) = 150

d = 4

c(4) + (-6c) = 150

4c - 6c = 150

-2c = 150

c = -75

Length: 2.6 meters

Width: 1.8 meters

Area: 4.68 meters

Since the scale is 1/40, we can multiply the centimeter measurements by 40 thus making it our scale factor.

Length

2.6 x 40 = 260

260 cm = 2.6 m

Width

4.5 x 40 = 180

180 cm= 1.8 m

Area

1.8 x 2.6 = 4.68 m

Length: 2.6 meters

Width: 1.8 meters

Area: 4.68 meters

Grandma should be able to make three slices of French toast by slicing one of the original slices into two.

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Answer: Option (3)

Step-by-step explanation:

[tex]\frac{1}{6a^2}-\frac{5b}{3a^3}+\frac{b^2}{4a}\\\\=\frac{2a}{12a^3}-\frac{20b}{12a^3}+\frac{3a^2 b^2}{12a^3}\\\\=\frac{3a^2 +b2 +2a-20b}{12a^3}[/tex]

Step-by-step explanation:

68% of all values are within 1 SD (110 ± 15).

95% of all values are within 2SD (110 ± 2×15 = 30)

99.7% of all values are within 3 SD (110 ± 3×15 = 45)

a)

95 is 110 - 15, so exactly 1 SD apart from the mean value.

68% or 0.68 is the probability of all values between 95 and 125.

so, 34% or 0.34 is the probability of all values between 95 and 110. and 50% or 0.5 is the probability of all values larger than 110.

so, the probability of all values larger than 95 is

0.34 + 0.5 = 0.84

b)

125 is 110 + 15, so exactly 1 SD apart from the mean value.

so, as per a) 34% or 0.34 is the probability of all values between 110 and 125. and 50% or 0.5 is the probability of all values larger than 110.

so, the probability of all values larger than 125 is

0.5 - 0.34 = 0.16