Consider a differentiable function f having domain all positive real numbers, and for which it is known that f'(x) = (4 - x)x^-3 for x > 0. (a) Find the t-coordinate of the critical point of f. Determine whether the point is a relative maximum, a relative minimum, or neither for the function f. Justify your answer. (b) Find all intervals on which the graph of f is concave down. Justify your answer. (c) Given that f(1) = 2, determine the function f.

The dimensions for the base and height that will make the tank weigh as little as possible are 10 feet and 5 feet, respectively

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

Volume = 500 cubic feet

Base = square base

This means that the volume can be calculated using

V = b²h

Where

b = base length

h = height

So, we have

b²h = 500

Make h the subject

h = 500/b²

The surface area is calculated as

A = 4bh + b²

Substitute h = 500/b² in A = 4bh + b²

A = 4b(500/b²) + b²

Evaluate

A = 2000/b + b²

Differentiate and set to 0

-2000/b² + 2b = 0

So, we have

2b = 2000/b²

Divide by 2

b = 1000/b²

Multiply by b²

b³ = 1000

So, we have

b = 10

Substitute b = 10 in h = 500/b²

h = 500/10²

Evaluate

h = 5

Hence, the height is 5 feet

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Inverse of the equation y = [tex]2x^2-8[/tex] is [tex]y = \pm \sqrt{\frac{x+8}{2} }[/tex]

What do you mean by inverse function?

Let f and g be two functions. If

f(g(x)) = x and g(f(x)) = x,

g is the reciprocal of f, and f is the reciprocal of g.

Some functions don't have inverses Let f be a function.

If the horizontal line crosses the graph of f more than once, then f has no inverse.

If no horizontal line crosses the graph of f more than once, then f is the inverse.

Given equation:

y = [tex]2x^2-8[/tex]

To find the inverse of the function find the value of the x in terms of y

[tex]2x^2-8=y[/tex]

[tex]2x^2=y+8[/tex]

[tex]x^{2} =\frac{y+8}{2}[/tex]

x = [tex]\pm \sqrt{\frac{y+8}{2} }[/tex]

Therefore, inverse of the equation y = [tex]2x^2-8[/tex] is [tex]y = \pm \sqrt{\frac{x+8}{2} }[/tex]

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The value of x = 2nπ,  (n ∈ Z) and

[tex]x=2(n\pi+(-1)^{n} \pi /6)[/tex],  (n ∈ Z)

What are Trigonometric Identities?

Trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation. Trigonometric Identities are true for every value of variables  being on both sides of an equation. Geometrically, these identities involve certain trigonometric functions(  similar as sine, cosine, tangent ) of one or  further angles.

According to question

⇒ sin(x/2) + cos(x) - 1 = 0

⇒ sin(x/2) + cos(x) - 1 = 0

{ cos(2x) = 1 - 2sin²(x) }

cos(x) = 1 - 2sin²(x/2)

⇒ sin(x/2) + cos(x) - 1 = 0

⇒ sin(x/2) + 1 - 2sin²(x/2) - 1 = 0

⇒ - 2sin²(x/2) + sin(x/2) + 1 - 1 = 0

⇒ - 2sin²(x/2) + sin(x/2)  = 0

⇒ sin(x/2)(- 2sin(x/2) + 1)  = 0

⇒ sin(x/2) = 0 , x = 2nπ (n ∈ Z)

and

⇒ sin(x/2) = 1/2 , [tex]x=2(n\pi+(-1)^{n} \pi /6)[/tex] (n ∈ Z)

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Answer: x = 70

Step-by-step explanation:

You can simply solve for x by simplifying both sides of the equation, then isolating the variable

Solve, x/7 - 2 = 8, for x.

x/7 = 10, added the value of 2 to both sides of the equation.

x=70, multiplied the value of 7 to both sides of the equation.

Thus the value of x is equal to 70.

Answer:

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

The salary during the 7th year is:

Total compensation through full 7 years is base salary for 7 years and total of the annual increase:

There is no property named 'left' for the circle. So the answer is 'c'.

Circle is a two dimensional figure which has a round shape. It is formed by joining all the points which are at an equal distance from a fixed point called the center of the circle.

The line formed by joining the center to any point on the boundary of the circle is called radius. Diameter is double of the radius.

Rotate angle is the term defining the angle of rotation of the circle. Angle of rotation of a complete circle is 2π.

Width is another term for diameter of the circle, although it is not commonly used.

Left is the only term which does not have any role in the properties of circles.

Hence, the term 'left' is not a valid property of circles.

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The equation that Eve can use to find the number of days d she can feed Pickles from the bag is; 4d = 24

The general formula for the equation of a line in slope intercept form is;

y = mx + c

where;

m is slope

c is y-intercept

We are given that;

Number of cups that Pickles eats per day = 4 cups per day

Now, Eve wants to know  how many days she can feed Pickles from a 24-cup bag of Hungry Hound.

If the number of days are denoted as d, then we have the equation as;

4d = 24

d = 24/4

d = 6 days

We conclude that she can feed the dog for 6 days.

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The function f(x) = 3[tex]x^{4}[/tex] - 5[tex]x^{3}[/tex] + 3 [tex]\sqrt{x^{2} +4}[/tex] is continuous at x = 2.

Given that,

We need to follow the following steps:

The function is:

f(x) = 3x^4 − 5x + ^3√(x^2 + 4)

The function is continuous at point x=2 if:

The function f(x) exists at x=2.

The limit on both sides of 2 exists.

The value of the function at x=2 is the same as the value of the limit of the function at x = 2.

Therefore:

The value of the function at x = 2 is:

f(x) = 3[tex]x^{4}[/tex] - 5[tex]x^{3}[/tex] + 3 [tex]\sqrt{x^{2} +4}[/tex]

f(2) = 3[tex](2^{4} )[/tex] - 5[tex](2^{3} )[/tex] + 3[tex]\sqrt{2^{2} +4}[/tex]

f(2) = 3*16 - 5*8 + 3 [tex]\sqrt{8}[/tex]

f(2) = 48 - 40 + 3*2.82

f(2) = 8 + 8.46

f(2) = 16.46

The limit of the f(x) is the same at both sides of x=2, that is, the evaluation of the limit for values coming below x = 2, or 1,0.5  is the same that the limit for values coming above x = 2, or 3 , 4 , 5 etc.

For this case:

[tex]lim_{x -2} f(x)[/tex] = 3[tex]x^{4}[/tex] - 5[tex]x^{3}[/tex] + 3 [tex]\sqrt{x^{2} +4}[/tex]

[tex]lim_{x -2} f(x)[/tex] = 16.46

Since

f(2) = 16.46

And

[tex]lim_{x -2} f(x)[/tex] = 16.46

Therefore,

The function f(x) = 3[tex]x^{4}[/tex] - 5[tex]x^{3}[/tex] + 3 [tex]\sqrt{x^{2} +4}[/tex] is continuous at x = 2.

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The number of square feet each lot measure is 54, 450 square feet.

Five acres of land were divided into 4 lots of equal size. The number of square feet each lot measure can be calculated as follows:

1 acre = 43,560 square ft

5 acres = ?

cross multiply

number of acres = 5 × 43,560

number of acres = 217800 square feet.

Therefore, the lots is divide into 4 equal lots.

Hence,

measure of each lot =  217800 / 4

measure of each lot = 54, 450 square feet.

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After 81 minutes of driving the distance to destination is 32.8 miles.

Th function that has two variables with first order term is called linear function.  While plotting on the XY coordinate system we get a straight line.

Why do we use linear equation in this problem?

as per question, distance to destination is a linear function of total driving time so we use the equation of straight-line having slope m and y intercept b for determining unknown values.

given, distance to destination in miles is a linear function of total driving time in minutes.

y=mx+b

in the equation, y denotes the distance to destination in miles

                 x is the total driving time in minutes

         m is the slope of linear equation

      b is the y intercept

Milan has 68 miles to his destination after 37 minutes of driving

68 = mx37+b

he has 45.6 miles to destination after 65 minutes of driving

45.6=mX65+b

solving the two equations by addition method

slope, m= -0.8

from the above two equation

we get b=97.6

now, after 81 minutes of driving the distance to destination, y=mx+b

                                               y= -0.8x81+97.6

                               distance = 32.8 miles

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The number of djembe drums you sold is 8

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

Total = 12 sets of maracas, 6 sets of claves, and x djembe drums.

Earning = $326

Also, we have:

The maracas costs $14 per set, the claves costs $5 per set, and the drum cost $16.

So, we have

12 * 14 + 6 * 5 + x * 16 = 326

Evaluate the products

198 + 16x = 326

Evaluate the iike terms

16x = 128

So, we have

x = 8

Hence, the number of djembe drums is 8

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Complete question

You sell instruments at a Caribbean music festival. You earn $326 by selling 12 sets of maracas, 6 sets of claves, and x djembe drums.

The maracas costs $14 per set, the claves costs $5 per set, and the drum cost $16.

Find the number of djembe drums you sold.

There are 21 days for there to be 150 frogs in the pond.

There are 37 days for there to be 1400 frogs in the pond.

What is exponential growth?

Quantity increases over time through a process called exponential growth. When a quantity's instantaneous rate of change with respect to time is proportional to the quantity itself, it happens.

Given:

There are currently 25 frogs in a (large) pond. The frog population grows exponentially, tripling every 10 days.

The exponential equation for the given problem is,

[tex]A(t) = Ar^t^/^1^0[/tex]

To find the number of days for there to be 150 frogs in the pond.

Here,

A(t) = 250, A = 25, r = 3

⇒

[tex]250 = 25(3)^t^/^1^0\\10 = 3^t^/^1^0\\t = 20.97[/tex]

t ≈ 21

Hence, there are 21 days for there to be 150 frogs in the pond.

Now to find how long for there to be 1400 frogs in the pond, we solve:

[tex]1400 = 25(3)^t^/^1^0\\56 = (3)^t^/^1^0\\t = 36.64[/tex]

t ≈ 37

Hence, there are 37 days for there to be 1400 frogs in the pond.

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Answer:

[tex]\frac{7}{10}[/tex]

Step-by-step explanation:

Answer:

Here is the answer

Step-by-step explanation:

If 40% of the students in the class are female, then there are 40/100 * 120 = 48 female students.

If 70 male students passed with a 100% pass rate, then the pass rate for male students is 100%.

If the pass rate for male students is 100%, then the pass rate for female students must also be 100%. This is because the pass rate is calculated as the number of students who pass divided by the total number of students, and if all the male students pass, then the pass rate for male students must be 100%.

Therefore, the pass rate for female students at a 100% rate is also 100%.

The expression equivalent to "20 divided by 1 plus 4" is 20 ÷ 1 + 4.

The rules of PEDMAS can be understood by the full form of the name. B stands for bracket, O for of, D for division, M for multiplication, A for addition and S stands for subtraction. For a mathematical expression involving the combination of any of these operators the priority will be given to the letter that comes first.

The given statement for the problem is, " 20 divided by 1 plus 4".

It can be written in the form of expression as follows,

The sign of plus is given as '+'.

And, for divide it is '÷'.

Now, 20 divided by 1 plus 4 can be written as,

20 ÷ 1 + 4

Which is evaluated  by BODMAS to give,

20 + 4 = 24.

Hence, the expression for the given case is 20 ÷ 1 + 4 which is equal to 24.

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Answer:

  (c)  x = -1/11(y -3)²

Step-by-step explanation:

You want an equation of the parabola with directrix x = 11/4 and vertex (0, 3).

The directrix is a vertical line to the right of the vertex. This means the parabola opens to the left.

The vertex form of the equation is ...

  x = 1/(4p)(y -k)² +h . . . . . . . . where (h, k) is the vertex, and p is the distance from the directrix to the vertex (also, the distance from the vertex to the focus)

Here, we have ...

  p = 0 -11/4 = -11/4 . . . . negative because the vertex is left of the directrix

and

  (h, k) = (0, 3)

so the equation is ...

  x = 1/(4(-11/4))(y -3)² +0

  x = -1/11(y -3)²

Answer:

Step-by-step explanation:

Given a total of 244 sodas and hot dogs sold, with sodas numbering 42 more than hot dogs, you want to know the number of each that were sold.

Let s represent the number of sodas sold. Then s-42 is the number of hot dogs sold, and the total sold is ...

  s +(s -42) = 244

Add 42 and simplify

  2s = 286

  s = 143 . . . . . divide by 2

  hot dogs = 143 -42 = 101 . . . . . figure the number of hot dogs

143 sodas and 101 hot dogs were sold.

After the markup in the price, the new cost of goldfish will be equal to $4.45.

The Latin phrase "per centum," which means "by the hundred," is where the English word "percentage" comes from. Percentage segments are those with a numerator of 100. In other words, it is a connection where the whole is always deemed to be valued 100.

As per the given information in the question,

Cost of goldfish = $3.45

Markup = 29%

So, the price increase will be,

(3.45 × 29)/100

= 100.05/100

= 1.0005

So, the new price is,

$3.45 + $1.0005

= $4.4505 ≈ $4.45

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Answer: 2 complete games

Step-by-step explanation:

add shoes and game charge

$10.00 + 2.75 = 12.75 <— that’s 1 complete game

subtract it from $30.00

$30.00 - 12.75 = 17.25.

A complete game costs 12.75. so, Kim had enough money for one more game.

17.25 - 12.75 = $4.50 remaining after 2 complete games

Answer: 7 games

Step-by-step explanation: if it charges 10 for shoes, then you already subtract 10 from 30. 30-10=20. then, to find out how many games she can play, divide 2.75 from 20. 20/2.75=7.272727.... Since you can't pay for half of a game, it would be 7 games.

The value of the expression is 3st² - s² is 315.

What is an expression?

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows: Expression: (Math Operator, Number/Variable, Math Operator)

Given expression is 3st² - s².

For s = 3 and t = 6, find the value of the expression 3st² - s².

To find the value of the expression, replace s by 3 and t by 6:

3st² - s²

= 3×3×6² - 3²

= (9 × 36) - 9

= 324 - 9

= 315

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44 seconds will pass before the planes are at the same altitude.

What do you mean by algebraic expression?

An algebraic expression in mathematics is an expression that consists of variables and constants and algebraic operations (addition, subtraction, etc.). Expressions are made up of concepts.

There are three main types of algebraic expressions:

Monomial Expression

Binomial Expression

Polynomial Expression

An algebraic expression with only one term is called a monomial.

It is given that plane A is at an altitude of 3500 feet and plane B is at an altitude of 2609 feet. Plane A is gaining altitude at 45.5 feet per second and plane B is gaining 65.75 feet per second.

Let x seconds will pass before the planes are at the same altitude.

Plane A, 3500 + 45.5x

Plane B, 2609 + 65.75x

x is number of seconds of time

When altitudes equal

3500 + 45.5x = 2609 + 65.75x

3500 - 2609 = 65.75x - 45.5x

891 = 20.25x

891/20.25 = x

44 = x

Therefore, 44 seconds will pass before the planes are at the same altitude.

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Answer:

y=-2x/3

Step-by-step explanation:

Perpendicular lines have opposite reciprocal slopes.

An example of opposite reciprocals: -1/2 and 2

-1*2=-2

The reciprocal of -2 is -1/2.

In order to solve for slope, isolate y:

3x - 2y = 7  ==> solve for y

3x = 7 + 2y  ==> add 2y on both sides to make y positive

3x - 7 = 2y  ==> isolate y by subtracting 7 on both sides

y = (3x - 7)/2  ==> divide both sides by 2 in order to isolate y

y = 3x/2 - 7/2  ==> distribution

Hence, the slope is 3/2  <== 3x/2.

Now, find the perpendicular slope:

-1*3/2=-3/2

The reciprocal of -3/2 is -2/3 ==> the slope of the perpendicular equation

Now, find the equation of the line that passes through (3, -2):

y=mx+b  ==> slope-intercept equation

-2=-2/3 * (3) + b  ==> plugin (x=3, y=-2) and m=-2/3 which is the slope.

-2 = 3 * (-2/3) + b  ==> solve for b

-2 = -6/3 + b

-2 = -2 + b

b = 0 ==> -2 + 0 = -2

y=-2x/3+0  ==> plugin the slope -2/3 and b=0.

y=-2x/3+0  ==> y=-2x/3

Hence, the equation is y=-2x/3.

Answer:

13

Step-by-step explanation:

Divide both sides of the equation by 4.

Answer:

The two locker numbers are 80 and 69.

Hope it helps!

Answer:

4/15 cups of soda per hour

Step-by-step explanation:

1 1/4 = 5/4 and if 5/4 hours yielded 1/3 cups, you would have to multiply 1/3 by the reciprocal to find the hourly rate

1         4             4

-   *     -      =      --

3        5             15

40 miles will he have after 85 minutes of driving .

Given :

austin is driving to san francisco, suppose that the distance to his destination in miles is a linear function of total driving time. has 67 miles to his destination after 49 minutes of driving . He has 47.5 miles to his destination after 75 minutes of driving .

slope = y2  - y1  / x2 - x1

m = 47.5 - 67 / 75 - 49

= -19.5 / 26

= -3/4

= -0.75

Use the point slope y - y1 = m(x - x1)

y - 67 = -.75 ( x - 49 )

y - 67 = -.75x + 36.75

y = -.75x + 36.75 + 67

y = -.75x + 103.75

x = 85

y = -.75(85) + 103.75

y = -63.75 + 103.75

y = 40 miles  

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Full question :

austin is driving to san francisco, suppose that the distance to his destnation in miles is a linear function of total driving time  has 67 miles to his destination after 49 minutes of driving . He has 47.5 miles to his destination after 75 minutes of driving . How many miles will he have after 85 minutes of driving ?

Given that x is a normal variable with mean = 47 and standard deviation = 6.4, find the following probabilities.

Generally, To solve these problems, we can use the standard normal distribution table or a computer program to find the desired probabilities.

First, we need to standardize the given values of x by subtracting the mean and dividing by the standard deviation. This gives us a standard normal random variable, denoted by Z, with a mean of 0 and a standard deviation of 1.

(a) P(x ≤ 60) = P(Z ≤ (60 - 47)/6.4) = P(Z ≤ 1.5625). From the standard normal distribution table or a computer program, we find that P(Z ≤ 1.5625) = 0.9356.

(b) P(x ≥ 50) = P(Z ≥ (50 - 47)/6.4) = P(Z ≥ 0.46875). From the standard normal distribution table or a computer program, we find that P(Z ≥ 0.46875) = 0.6744.

(c) P(50 ≤ x ≤ 60) = P(0.46875 ≤ Z ≤ 1.5625) = P(Z ≤ 1.5625) - P(Z ≤ 0.46875) = 0.9356 - 0.6744 = 0.2612.

Therefore, the probabilities are as follows:

(a) P(x ≤ 60) = 0.9356 (b) P(x ≥ 50) = 0.6744 (c) P(50 ≤ x ≤ 60) = 0.2612

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Answer:

Below

Step-by-step explanation:

2^9 possibilities =512

9 possibles with only ONE tails  (9 C 1)

36 possibles with TWO tails     ( 9 C 2)

45 out of 512       45/512 = .088  

The required equation of the line which contains the points (−7,2) and (2,−2) is y = -4/9x - 8/9.

A straight line's general equation is y = MX + c, where m denotes the gradient and y = c denotes the point at which the line crosses the y-axis. On the y-axis, this value c is referred to as the intercept.

The equation of the line's general form is:

y = MX + c

Where, m = slope, c = y-intercept and slope = ( y₂ - y₁ ) / ( x₂ - x₁ ).

Determine the line's slope:

Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )

Slope = ( -2-2) / (2+7 )

Slope= -4/9

The y-intercept is determined as follows:

y = MX + c

2 = -4/9x 2+c

c = -8/9

The lines' equation:

y = MX + c

y = -4/9x - 8/9

Therefore, the required equation of the line which contains the points (−7,2) and (2,−2) is y = -4/9x - 8/9.

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