(1 point) Use the Integral Test to determine whether the infinite series is convergent. 8W7 n 5 n=1 Fill in the corresponding integrand and the value of the improper integral. Enter inf for , -inf for -oo, and DNE if the limit does not exist. - Compare with Soo dx = By the Integral Test, the infinite series Σ -5 п n=1 O A. converges B. diverges Note: You can earn partial credit on this problem.

Using the fact that the sum of the angles in a triangle is 180 degrees we do know that their sum is 119 degrees.

To understand why we used the fact that the sum of angles in a triangle is 180 degrees, let's take a closer look at the diagram.

We see that Angle 2 and Angle 5 are on the same side of a transversal and are therefore supplementary angles. This means that their sum is 180 degrees:

Angle 2 + Angle 5 = 180

We can substitute the value of Angle 5 (49 degrees) for Angle 5 and x + 12 for Angle 2 to get:

x + 12 + 49 = 180

Simplifying the equation, we get:

x = 119 - 49 - 12

x = 58

This gives us the value of x, but not the measure of Angle 2. To find the measure of Angle 2, we need to use the fact that the sum of angles in a triangle is 180 degrees.

We can write an equation using angles 2, 3, and 4 (which we now know is 49 degrees) as follows:

Angle 2 + Angle 3 + Angle 4 = 180

Substituting the known values, we get:

x + 12 + Angle 3 + 49 = 180

Simplifying the equation, we get:

x + Angle 3 = 119

So, we know that the sum of Angle 3 and x is 119 degrees, but we still don't know the measure of either angle on its own.

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The divergence of F is 2y cos(z) + eˣ cos(z) + 3xe³, and the curl of F is -eˣcos(z)i - 3xe³j + (eˣsin(z) + 2cos(z))k.

To find the divergence and curl of the vector field F(x, y, z) = 2y cos(z)i + eˣ sin(z)j + xe³k, we need to apply the appropriate operators.

The divergence of F is given by:

div F = ∇ · F = (∂/∂x)i + (∂/∂y)j + (∂/∂z)k · (2y cos(z)i + eˣ sin(z)j + xe³k)

where ∇ is the del operator.

Calculating the dot product, we get:

div F = 2y cos(z) + eˣ cos(z) + 3xe³

Therefore, the divergence of F is:

div F = 2y cos(z) + eˣ cos(z) + 3xe³

Now, let's find the curl of F. The curl of F is given by:

curl F = ∇ × F = ( (∂/∂y)(xe³) - (∂/∂z)(eˣsin(z)) )i - ( (∂/∂x)(2ycos(z)) - (∂/∂z)(xe³) )j + ( (∂/∂x)(eˣsin(z)) - (∂/∂y)(2ycos(z)) )k

Calculating the partial derivatives, we get:

(∂/∂y)(xe³) = 0

(∂/∂z)(eˣsin(z)) = eˣcos(z)

(∂/∂x)(2ycos(z)) = 0

(∂/∂z)(xe³) = 3xe³

(∂/∂x)(eˣsin(z)) = eˣsin(z)

(∂/∂y)(2ycos(z)) = -2cos(z)

Substituting these values, we get:

curl F = (0 - eˣcos(z))i - (0 - 3xe³)j + (eˣsin(z) - (-2cos(z)))k

Simplifying, we get:

curl F = -eˣcos(z)i - 3xe³j + (eˣsin(z) + 2cos(z))k

Therefore, the divergence of F is 2y cos(z) + eˣ cos(z) + 3xe³, and the curl of F is -eˣcos(z)i - 3xe³j + (eˣsin(z) + 2cos(z))k.

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According to the information the length of the ladder to reach the hoop will be approximately 10.77 feet.

How to calculate the length of the ladder?

 Analyzing the problem, we can see that the ladder, the height and the distance from the base of the basket will form a right triangle. We can then use the Pythagorean theorem to calculate the length of the ladder, which will be the hypotenuse of the triangle. The formula used will be:

Substituting the information in the formula we have:

So let's use the square root of 116 to find how long the ladder must be to reach the hoop, which in this case will be:

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The minimum force required to move a block of 35 N resting on a steel table with a coefficient of static friction of 0.40 is 14 N.

Friction refers to the force that resists the motion and thus the force acts in the opposite direction of the force applied.

There are the following types of friction:

1. Static Friction

2. Limiting Friction

3. Kinetic Friction

F = μN

where μ is the coefficient of friction

N is the Normal Force

When the object is resting on a table, Normal force is the weight.

N = 35 N

μ = 0.40

F = 0.4 * 35

= 14 N

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Answer:

x = 4

Step-by-step explanation:

10 - 4x + 6 - 2x = -2x

10 - 6x + 6 = -2x

16 - 6x = -2x

16 - 4x = 0

-4x = -16

x = 4

Answer:

x = 4

Step-by-step explanation:

Add like terms

-6x + 16 = -2x

Bring like terms to the opposite side

16 = 4x

Divide both sides by 4

x = 4

Answer:

parabola

Step-by-step explanation:

The graph shape is a parabola, opens-up type

Mr. Agber needs to employ 48 workers to cultivate his 8 acres of farmland in 6 days. The exact relationship between the parameters is W × D = K × L, and the constant (K) in this case is 36.

To solve this problem, we can use the concept of direct variation. The relationship between the number of workers, the size of the land, and the number of days can be expressed as follows:

Number of Workers (W) × Number of Days (D) = Constant (K) × Size of the Land (L)

In Mr. Agber's case, we know the initial situation is:

20 workers × 9 days = K × 5 acres

To find the constant, K, we can rearrange the equation:

K = (20 workers × 9 days) / 5 acres
K = 180 / 5
K = 36

Now that we have the constant, we can use it to determine the number of workers needed for the 8 acres of land in 6 days:

W × 6 days = 36 × 8 acres

Again, rearrange the equation to find the number of workers, W:

W = (36 × 8 acres) / 6 days
W = 288 / 6
W = 48 workers

So, Mr. Agber needs to employ 48 workers to cultivate his 8 acres of farmland in 6 days. The exact relationship between the parameters is W × D = K × L, and the constant (K) in this case is 36.

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The answer is (d) –0.9568, –0.9659.

To find the first five terms of the trigonometric series for cos(4π/7), we can use the formula:

cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...

Substituting x = 4π/7, we get:

cos(4π/7) = 1 - (4π/7)²/2! + (4π/7)⁴/4! - (4π/7)⁶/6! + (4π/7)⁸/8!

Using a calculator to evaluate each term and rounding to four decimal places, we get:

cos(4π/7) ≈ -0.9568

Comparing this approximation to the actual value of cos(4π/7), which is approximately -0.9659, we see that the approximation is fairly close but not exact. So, the answer is (d) –0.9568, –0.9659.

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The expression that is equivalent to the one Regina wrote is y + 27/4

From the question, we have the following parameters that can be used in our computation:

y + 9 x 3/4

This means that

Expression = y + 9 x 3/4

Expanding the above expression, we have

Expanded expression = y + 27/4

Using the above as a guide, we have the following:

The expression that is equivalent to the one Regina wrote is y + 27/4

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[tex]$x+3= \pm \sqrt{\frac{21}{2}}$[/tex]  Thus, option D is correct.

the quadratic equation  [tex]2x^2+12x-3=0$ is \ $x = \frac{-6 \pm \sqrt{42}}{2}[/tex], which simplifies to [tex]$x = -3 \pm \frac{\sqrt{42}}{2}$.[/tex]

However, the three options listed are the steps that Inga could use to solve the quadratic equation, and only three of them are correct. The correct options are:

[tex]$2\left(x^2+6 x\right)=-3$[/tex]

[tex]$2\left(x^2+6 x\right)=3$[/tex]

[tex]$x+3= \pm \sqrt{\frac{21}{2}}$[/tex]

Option 1 is the result of dividing both sides of the original equation by 2, which simplifies the coefficients.

Option 2 is the result of adding $\frac{3}{2}$ to both sides of the equation to isolate the quadratic terms. Option 3 is the final step, where the equation is solved for $x$ by completing the square and taking the square root of both sides.

Therefore,  it is not one of the three steps that Inga could use to solve the quadratic equation. [tex]$x+3= \pm \sqrt{\frac{21}{2}}$[/tex]

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Inga is solving [tex]$2 x^2+12 x-3=0$.[/tex] Which steps could she use to solve the quadratic equation? Select three options.

[tex]$2\left(x^2+6 x+9\right)=3+18$[/tex]

[tex]$2\left(x^2+6 x\right)=-3$[/tex]

[tex]$2\left(x^2+6 x\right)=3$[/tex]

[tex]$x+3= \pm \sqrt{\frac{21}{2}}$[/tex]

By using  sum , difference , product , quotient and one that involves at least two operations that have the value of -3/4.

Now, We have to find the five expressions :

Expression for sum is -

1/4 + (-1) = -3/4

Expression for difference is -

(1/4 - 1) = -3/4

Expression for product is -

(-3/2)(1/2) = -3/4

Expression for quotient is -

(1 - 4)/4 = -3/4

Expression that involves at least two operations is -

-(1/4 + 2/4) = -3/4

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The area of a sector bounded by a 180° arc with a radius of 8 centimeters is 32π square centimeters in simplest form.

To find the area of a sector bounded by a 180° arc with a radius of 8 centimeters, you can follow these steps:

Step 1: Recall the formula for the area of a circle: A = πr², where A is the area and r is the radius.

Step 2: Calculate the area of the entire circle with a radius of 8 centimeters: A = π(8)² = 64π square centimeters.

Step 3: Determine the fraction of the circle represented by the 180° arc. Since a full circle is 360°, the fraction is 180°/360°, which simplifies to 1/2.

Step 4: Multiply the area of the entire circle by the fraction to find the area of the sector: (1/2) * (64π) = 32π square centimeters.

So, the area of a sector bounded by a 180° arc with a radius of 8 centimeters is 32π square centimeters in simplest form.

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The third approximation of the fourth root of 4, starting with x0 = 1, is approximately 1.7321

To find the third approximation, X2, of the positive fourth root of 4 using Newton's method, we will follow these steps:

1. Define the function f(x) = x^4 - 4 and its derivative f'(x) = 4x^3.
2. Start with an initial guess x0 = 1.
3. Apply Newton's method formula to find the next approximation: x1 = x0 - f(x0) / f'(x0).
4. Repeat the process for the second and third approximations.

Step 1:
f(x) = x^4 - 4
f'(x) = 4x^3

Step 2:
x0 = 1

Step 3:
x1 = x0 - f(x0) / f'(x0) = 1 - (1^4 - 4) / (4 * 1^3) = 1 - (-3 / 4) = 1 + 0.75 = 1.75

Step 4:
x2 = x1 - f(x1) / f'(x1) = 1.75 - (1.75^4 - 4) / (4 * 1.75^3) ≈ 1.7321

The third approximation of the fourth root of 4 determined by calculating the third approximation of the right of f(x) = x^4 - 4, starting with x0 = 1, is approximately 1.7321 (rounded to four decimal places).

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To find the first term and common ratio of a geometric sequence, we can use the formula for the nth term:

a_n = a_1 * r^(n-1)

We are given the eighth and eleventh terms, so we can set up two equations:

a_8 = a_1 * r^(8-1) = 81920

a_11 = a_1 * r^(11-1) = 5242880

After dividing the second with by the first equation, we get:

(a_1 * r^(11-1)) / (a_1 * r^(8-1)) = 5242880 / 81920

Simplifying, we get:

r³ = 64

Doing the root of cube both sides, we get:

r = 4

Substituting this into the first equation, we get:

a_1 * 4^(8-1) = 81920

a_1 * 4^7 = 81920

a_1 = 5

Therefore, the first term is 5 and the common ratio is 4.

In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor called the common ratio (r). The formula for the nth term of a geometric sequence is:

an = a * r^(n-1)

Given that the 8th term (a8) is 81,920 and the 11th term (a11) is 5,242,880, we can set up the following equations:

81920 = a * r^(8-1) => 81920 = a * r⁷ (1)

5242880 = a * r^(11-1) => 5242880 = a * r¹⁰ (2)

Now, we need to find the values of a (the first term) and r (the common ratio). Divide equation (2) by equation (1):

(5242880 / 81920) = (a * r¹⁰) / (a * r⁷)

64 = r^3

Now, we can find the common ratio r:

r = 4 (since 4³ = 64)

Next, substitute r back into equation (1) to find the first term a:

81920 = a * 4⁷

a = 81920 / 16384

a = 5

So, the first term (a) is 5, and the common ratio (r) is 4.

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The relation a into b represent  all possible combinations of elements in a and b as ordered pairs.

The relation "a into b" refers to the cartesian product of the sets a and b, which is denoted by a × b. The cartesian product of two sets is a set of all possible ordered pairs, where the first element of each ordered pair comes from the first set, and the second element comes from the second set.

In this case, a = {0, 1, 2} and b = {-1, 1, 2}. So, their cartesian product a × b is the set of all possible ordered pairs (a, b), where a is an element of a and b is an element of b. Therefore, we have:

a × b = {(0, -1), (0, 1), (0, 2), (1, -1), (1, 1), (1, 2), (2, -1), (2, 1), (2, 2)}

This means that the relation a into b consists of all possible combinations of elements in a and b as ordered pairs.

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Answer:

Step-by-step explanation:

First, we can simplify the expression inside the parentheses by combining like terms:

-2(x + 12y - 5 - 17x - 16y + 4) = -2(-16x - 4y - 1)

Next, we can distribute the -2 to each term inside the parentheses:

-2(-16x - 4y - 1) = 32x + 8y + 2

Therefore, -2(x + 12y - 5 - 17x - 16y + 4) simplified is 32x + 8y + 2.

The simplified expression is 32x + 8y + 2.

Simplification of an algebrai expression can be defined as the process of writing an expression in the most efficient and compact form without affecting the value of the original expression.

The process entails collecting like terms, which implies adding or subtracting terms in an expression.

Simplify the expression -2(x + 12y - 5 - 17x - 16y + 4).

First, let's distribute the -2 to each term inside the parentheses:

-2(x) + (-2)(12y) - (-2)(5) - (-2)(17x) - (-2)(16y) + (-2)(4)

Now we'll multiply: -2x - 24y + 10 + 34x + 32y - 8

Next, we'll combine like terms:

(-2x + 34x) + (-24y + 32y) + (10 - 8)

The result is 32x + 8y + 2

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C. The constant of proportionality between the diameter and circumference of a circle is pi.

The constant pi comes from the relationship between the diameter and circumference of a circle.

Constant of Proportionality

A constant of proportionality is a number that describes the ratio between 2 values. No matter the measurements of a circle, the constant of proportionality between a circumference and diameter is always the same. This means that the circumference divided by the diameter ≈ 3.14.

Pi

Pi is an irrational number that can be estimated but never completely solved. The value of pi can be used to complete many different calculations such as the area of a circle, and it is used in many different functions like sin. For this reason, pi is one of the most important constants in math.

Answer:

It will cost $288 to carpet the living room at $36 per square yard.

Step-by-step explanation:

First, we need to convert the room dimensions to square yards, since the carpet price is given in square yards.

The area of the living room is:

[tex]\sf:\implies 6\: ft \times 12\: ft = 72\: ft^2[/tex]

To convert this to square yards, we divide by 9 (since there are 9 square feet in a square yard):

[tex]\sf:\implies \dfrac{72\: ft^2}{9} = 8\: yards^2[/tex]

So the living room is 8 square yards in area.

To find the cost of carpeting the room, we multiply the area by the cost per square yard:

[tex]\sf:\implies 8\: yards^2 \times \$36/square\: yard = \boxed{\bold{\:\:\$288\:\:}}\:\:\:\green{\checkmark}[/tex]

Therefore, it will cost $288 to carpet the living room at $36 per square yard.

Jenny needs 180g of butter, 210g of brown sugar, 375g of oats, and 3 tablespoons of syrup to make 24 flapjacks.

To make 24 flapjacks, Jenny needs to increase the amount of each ingredient proportionally.

To calculate the required amounts, we can use ratios. If 16 flapjacks require 120g of butter, then 24 flapjacks require:

Butter: (24/16) x 120g = 180g

Brown sugar: (24/16) x 140g = 210g

Oats: (24/16) x 250g = 375g

Syrup: (24/16) x 2 tablespoons = 3 tablespoons

Therefore, Jenny needs 180g of butter, 210g of brown sugar, 375g of oats, and 3 tablespoons of syrup to make 24 flapjacks.

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The correct answer is (D) 2,205.

To calculate the number of different combinations of tag codes, we need to consider the possible options for each position in the code.

For the first and third positions (consonants), we have 21 options for each position since there are 21 consonants in the English language.

For the middle position (vowel), we have 5 options since there are 5 vowels in the English language.

Therefore, the total number of different combinations is calculated by multiplying the number of options for each position:

21 (consonant options) * 5 (vowel options) * 21 (consonant options) = 2,205

Therefore, the correct answer is (D) 2,205.

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Answer:

The answer is A. 135 cm2.

Area of a parallelogram is base * height. In this case, the base is 24 cm and the height is 9 cm. Therefore, the area is 24 * 9 = 135 cm2.

Answer: the answer would be 360

Step-by-step explanation:

the equation for area of a parallelogram is base x height.

The base is 24 as it is at the top of the shape.

The height is 15 as well since a parallelogram is congruent.

multiply the two and it gives you 360

Probability of a student participating in football: 0.4 or 40%

Probability of a student participating in tennis: 0.2 or 20%

Assuming that the table lists the number of students who participate in each sport out of a total of 150 students, we can find the probability of a student participating in each sport by dividing the number of students who participate in each sport by the total number of students:

Probability of a student participating in football:

Number of students who participate in football / Total number of students = P(Football)

Probability of a student participating in tennis:

Number of students who participate in tennis / Total number of students = P(Tennis)

For example, if the table shows that 60 students participate in football and 30 students participate in tennis out of a total of 150 students, then the probabilities would be:

Probability of a student participating in football:

60/150 = 0.4 or 40%

Probability of a student participating in tennis:

30/150 = 0.2 or 20%

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The discount price of the magazine was $2.30.

The convenience store purchased the magazine at an original cost of $2.30 and marked it up 100%. Find the selling price after the markup as follows.

1. Calculate the markup amount:

100% of $2.30 (Original cost * Markup percentage)

Markup amount = $2.30 * 100% = $2.30

2. Add the markup amount to the original cost to get the selling price.

Selling price = Original cost + Markup amount = $2.30 + $2.30 = $4.60

Next, the store placed the magazine on sale for 50% off.

3. Calculate the discount amount:

50% of the selling price (Selling price * Discount percentage)

1. Discount amount = $4.60 * 50% = $2.30

4. Subtract the discount amount from the selling price to get the discount price.

Discount price = Selling price - Discount amount = $4.60 - $2.30 = $2.30

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the velocity vector of the object is v(2) = 3i + j + k, the speed is sqrt(11), and the acceleration vector is a(t) = (1/2)k.

We're given the position vector and time and asked to find the velocity vector, speed, and acceleration vector of the object. Let's solve this step-by-step.

1. Differentiate the position vector with respect to time (t) to find the velocity vector:
Position vector: r(t) = 3ti + tj + (1/4)t^2k

Velocity vector: v(t) = dr(t)/dt = d(3ti)/dt + d(tj)/dt + d((1/4)t^2k)/dt

v(t) = 3di/dt + dj/dt + (1/2)tk

v(t) = 3i + j + (1/2)tk

2. Plug in the given time (t = 2) into the velocity vector to find the velocity at that time:
v(2) = 3i + j + (1/2)(2)k

v(2) = 3i + j + k

3. Find the speed by calculating the magnitude of the velocity vector:
Speed = |v(2)| = sqrt((3^2) + (1^2) + (1^2))

Speed = sqrt(9 + 1 + 1)

Speed = sqrt(11)

4. Differentiate the velocity vector with respect to time (t) to find the acceleration vector:
Acceleration vector: a(t) = dv(t)/dt = d(3i)/dt + d(j)/dt + d((1/2)tk)/dt

a(t) = 0i + 0j + (1/2)k

So, the velocity vector of the object is v(2) = 3i + j + k, the speed is sqrt(11), and the acceleration vector is a(t) = (1/2)k.

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The probability of picking a 20 dollar bill is 1/7 or 14.3%

The probability of a thing is the likelihood or number of chances that such a thing will occur. For the scenario given,

There are a total of 7 bills in your pocket

1, 1, 1, 1,

5, 5,

20.

To find the probability of picking out the twenty dollar bill, divide the number of twenty dollar bills by the the total of the number of bills you are with.

Probability = 1/ 7 which can be converted to % = 14.3%.

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Answer:

24

Step-by-step explanation:

add all of them up

a) The profit at a sales level of 14 trees is $180.40.

b) The average change in profit sales is $13.96 per tree [(ΔP/Δx) = 97.75/7].

c) The instantaneous rate of change of profit at a sales level of 14 trees is $19.72 per tree.

d) The equation of the tangent line at (5, f(5)) is y = 6x - 25.

a. To find the profit at a sales level of 14 trees, we need to evaluate the profit function at x = 14:

P(x) = 20x - 0.01x^2 - 100

P(14) = 20(14) - 0.01(14)^2 - 100 = $180.40

Therefore, the profit at a sales level of 14 trees is $180.40.

b. To find the average change in profit sales from 12 to 19 trees, we need to calculate the average rate of change of the profit function over this interval:

Δx = 19 - 12 = 7

ΔP = P(19) - P(12) = (2019 - 0.0119^2 - 100) - (2012 - 0.0112^2 - 100) = $97.75

Therefore, the average change in profit sales is $13.96 per tree [(ΔP/Δx) = 97.75/7].

c. To find the instantaneous rate of change of profit at a sales level of 14 trees, we need to find the derivative of the profit function at x = 14:

P(x) = 20x - 0.01x^2 - 100

P'(x) = 20 - 0.02x

P'(14) = 20 - 0.02(14) = $19.72

Therefore, the instantaneous rate of change of profit at a sales level of 14 trees is $19.72 per tree.

d. To find the equation of the tangent line at (5, f(5)), we need to find the slope of the tangent line and its y-intercept:

f(x) = x^2 - 4x

f'(x) = 2x - 4

f'(5) = 2(5) - 4 = 6

The slope of the tangent line at (5, f(5)) is 6.

To find the y-intercept of the tangent line, we can use the point-slope form of a line:

y - f(5) = m(x - 5)

y - (5^2 - 4*5) = 6(x - 5)

y - 5 = 6x - 30

y = 6x - 25

Therefore, the equation of the tangent line at (5, f(5)) is y = 6x - 25.

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The calories she burns per minute is 17 cal

We are given that the total calories that are burnt in 15 minutes and 20 minutes are 255 and 340 respectively.

We can use the equation

total calorie burnt = time (in minutes) * calorie burnt in one minute

here we know the total calorie that is burnt and the time, we can substitute the calorie that is burnt in a single minute with 'n'.

we can say that :

255 = 15 * x

x=255/15

x= 17.

The total calorie burn per minute is 17.

now for the verification, we know that if the total calorie burn per minute is 17 it should satisfy both the equation.

So, 340 = 20*x

340=20* 17

340 = 340

Thus it satisfies both equations.

Hence the calorie burnt per minute = 17

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Answer: range=9 mode=3 mean=9

Step-by-step explanation:

The given system of equations has one solution.

To determine the number of solutions of the functions. The given system over the interval (-3, 1], we need to find the intersection points of the two functions, f(x) and g(x), within that interval.

First, let's analyze each function separately:

The natural logarithm function ln(x) is only defined for positive                  values of x. In this case, we have ln(x + 3). To find the intersection points              with the interval (-3, 1], we need to ensure that x + 3 is positive.

For x in the interval (-3, 1], we have:

-3 < x ≤ 1

Adding 3 to both sides of the inequality:

0 < x + 3 ≤ 4

Therefore, the function f(x) = ln(x + 3) is defined over the interval (0, 4].

    2. Function g(x) = 2 * [tex]6^x[/tex]:

The exponential function [tex]6^x[/tex] is always positive for any real value of x. Multiplying it by 2 won't change the fact that the function remains positive. Hence, g(x) is positive for all real values of x.

Now, let's determine the intersection points of f(x) and g(x) within the interval (-3, 1].

Since g(x) is always positive and f(x) is defined over (0, 4], the intersection points occur where f(x) = g(x) > 0.

To solve this equation, we can rewrite it as ln(x + 3) - 2 * [tex]6^x[/tex] = 0.

Finding the exact solutions to this equation is not straightforward and may require numerical methods or graphing. However, it's clear that there is at least one solution within the interval (0, 4].

In conclusion, the given system has at least one solution over the interval (-3, 1].

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