The volume of this prism is 2990cm3. the area of the cross-section is 65cm2. work out x

To find the height of the square pyramid, we will use the Pythagorean theorem. Given the area of the base is 64 cm² and the slant height is 9 cm, let's first find the side length of the base.

Since it's a square, the area of the base is side length squared (s²). Therefore, s² = 64 cm². Taking the square root of both sides, we get s = 8 cm.

Now, let the height be h and use the Pythagorean theorem with the side length (8 cm) and the slant height (9 cm):

h² + (s/2)² = (slant height)²
h² + (8/2)² = 9²
h² + 4² = 81
h² + 16 = 81
h² = 65

Taking the square root of both sides:

h = √65 cm ≈ 8.06 cm

The height of the pyramid is approximately 8.06 cm.

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The system of equations is.

x + y = 80

12x + 10y = 900

And the solutions are y = 30 and x = 50

Let's define the two variables:

x = number of calculators.

y = number of calendars.

With the given information we can write two equations, then the system will be:

x + y = 80

12x + 10y = 900

Now let's solve it.

We can isolate x on the first equation to get:

x = 80 - y

Replace that in the other equation to get:

12*(80 - y) + 10y = 900

-2y = 900 - 960

-2y = -60

y = -60/-2 = 30

Then x = 50

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For each function, all four second-order partial derivatives are f(x,y) are (x + y)2, (x + y), 3x^2y + 5xy^2, 2xy, (x + y)e^y, xe^y, sin(x/y), x^2 + y^2, 5x^3y^2 - 7xy^3 + 9x^2 +11, sin(x^2 + y^2) and 3 sin(2x) cos(5y). It is proved that f x y is equals to f y x.

f(x,y) = (x + y)2

f x x = 2, f xy = 2, f yx = 2, f y y = 2

Since f x y = fy x, the mixed partial derivatives are equal.

f(x,y) = (x + y)

f x x = 0, f x y = 1, f y x = 1, f y y = 0

Since f x y = f y x, the mixed partial derivatives are equal.

f(x, y) = 3x^2y + 5xy^2

f x x = 6y, f x y = 6x + 10y,  f y x = 6x + 10y, f y y = 10x

Since f x y = f y x, the mixed partial derivatives are equal.

f(x, y) = 2 x y

f x x = 0, f x y = 2, f y x = 2, f y y = 0

Since f x y = f y x, the mixed partial derivatives are equal.

f(x,y) = (x + y) * e^y

f x x = e^y, f x y = e^y + e^y, f y x = e^y + e^y, f y y = (x + 2y) * e^y

Since f x y = f y x, the mixed partial derivatives are equal.

f(x,y) = x * e^y

f x x = 0, f x y = e^y, fy x = e^y, f y y = x * e^y

Since fx y = fy x, the mixed partial derivatives are equal.

f(x, y) = sin(x/y)

f x x = -sin(x/y) / y^2, f x y = cos(x/y) / y^2,  f y x = cos(x/y) / y^2, f y y = -x * cos(x/y) / y^4 - sin(x/y) / y^2

Since f x  y = f y x, the mixed partial derivatives are equal.

f(x, y) = x^2 + y^2

f x x = 2, f x y = 0, f y x = 0, f y y = 2

Since f x y = f y x, the mixed partial derivatives are equal.

f(x, y) = 5x^2y^2 - 7xy + 9x^2 + 11

f x x = 10xy^2 + 18, f x y = 10x^2y - 7, f y x = 10x^2y - 7, fy y = 10x^2y^2

Since fx y = fy x, the mixed partial derivatives are equal.

f(x,y) = sin(x^2 + y^2)

fx x = 2xcos(x^2 + y^2), fx y = 2ycos(x^2 + y^2), fy x = 2ycos(x^2 + y^2), fy y = 2x * cos(x^2 + y^2)

Since fx y = fy x, the mixed partial derivatives are equal.

f(x,y) = 3sin(2x)cos(5y)

fx x = 0, fx y = -30sin(2x)sin(5y), fy x = -30sin(2x)sin(5y), fy y = 0

Since fx y = fy x, the mixed partial derivatives are equal.

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The account that will give Grady at least a 5% annual yield is Account C

We can use the formula for compound interest to compare the three investment accounts and find the one that will give Grady at least a 5% annual yield:

FV = PV × (1 + r/n)^(n*t)

where FV is the future value, PV is the present value, r is the annual interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the number of years.

For Account A:

APR = 4.95%, compounded monthly

r = 0.0495

n = 12

t = 1

FV = PV × (1 + r/n)^(nt)

FV = PV × (1 + 0.0495/12)^(121)

FV = PV × 1.050452

To get at least a 5% annual yield, we need FV/PV ≥ 1.05

1.050452/PV ≥ 1.05

PV ≤ 1.000497

Therefore, Account A will not give Grady at least a 5% annual yield.

For Account B:

APR = 4.85%, compounded quarterly

r = 0.0485

n = 4

t = 1

FV = PV × (1 + r/n)^(nt)

FV = PV × (1 + 0.0485/4)^(41)

FV = PV × 1.049375

To get at least a 5% annual yield, we need FV/PV ≥ 1.05

1.049375/PV ≥ 1.05

PV ≤ 1.000351

Therefore, Account B will not give Grady at least a 5% annual yield.

For Account C:

APR = 4.75%, compounded daily

r = 0.0475

n = 365

t = 1

FV = PV × (1 + r/n)^(nt)

FV = PV × (1 + 0.0475/365)^(3651)

FV = PV × 1.049038

To get at least a 5% annual yield, we need FV/PV ≥ 1.05

1.049038/PV ≥ 1.05

PV ≤ 1.000525

Therefore, Account C will give Grady at least a 5% annual yield.

Therefore, the account that will give Grady at least a 5% annual yield is Account C.

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Triangle ABD is scalene and obtuse.

The triangle has three sides of unequal length, hence it is classified as an scalene triangle.

(it would be equilateral if all had the same length, and isosceles if two sides have the same length).

The triangle has one angle larger than 90º, hence it is classified as an obtuse triangle.

(acute with no angles of 90º or greater, right with one angle of exactly 90º).

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Answer:

Around 3 years

Step-by-step explanation:

6% of 30000 is 1800. I subtracted 1800 from 30000 until it got down too 24000 so im assuming that's how old

The test statistic (or standard score) would be 1.72.

To convert the outcome of the pulse rate data to a standard score, we would need to calculate the z-score. The formula for the z-score is: (sample mean - population mean) / (standard deviation / square root of sample size).

In this case, the sample mean is 76, the population mean (according to the report) is 72, the standard deviation is 13, and the sample size is 85. Plugging these values into the formula, we get:

(76 - 72) / (13 / sqrt(85)) = 1.72.

Therefore, the test statistic (or standard score) is 1.72. This indicates that the sample mean of 76 is 1.72 standard deviations above the population mean of 72. This information can be used to conduct a statistical test of significance and determine whether the difference between the sample mean and population mean is statistically significant.

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The given expression is a trinomial because it consists of three terms: 4x²-y+oz⁴

A trinomial is a polynomial with three terms. It is a type of algebraic expression that consists of three monomials connected by addition or subtraction. The general form of a trinomial is:

ax^2 + bx + c

A monomial is an algebraic expression that consists of a single term. It is a polynomial with only one term. A term is a combination of a coefficient and one or more variables raised to non-negative integer exponents.  The general form of a monomial is:c * xᵃ,  yᵇ, zⁿ....

where 'c' represents the coefficient (a constant), and 'x', 'y', 'z', etc., represent variables, each raised to a non-negative exponent (a, b, n, etc.).

example of monomials:  5x² - This monomial has a coefficient of 5 and a single variable 'x' raised to the power of 2.

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The radar beam moves at a pace of roughly 536.47 kilometers per hour as it scans the coastline.

Let A represent the location of the radar antenna and B represent the shoreline location that is closest to A. Let C represent the radar beam's current location on the coast and Ф represent the angle between the beam and the line AB. As a result, we obtain a right triangle ABC, where AB is equal to 4 km, and BC is the length at which the radar beam sweeps along the shore.

32 rev/min(2π/60 sec) = 3.36 radians/sec. BC = r(Ф) = (4 km)(π/4) = π km.

We may calculate the radar beam's speed down the shore by multiplying these two values:

(536.47 km/hr) = 10.54 km/sec or (3.36 rad/sec)(π km).

Hence, the of sweeping is 10.54 km/sec.

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The actual percent loss on the original $7000 investment is 46%. This means that the investment consultant's statement of a 10% overall increase is not accurate.

To calculate the actual percent gain or loss on the original $7000 investment, we can use the following formula:

Actual percent gain or loss = (Ending value - Beginning value) / Beginning value * 100%

At the end of the first year, the investment decreased by 70% of its original value, which means its value was only 30% of $7000, or $2100.

At the end of the second year, the investment increased by 80% of the value it had at the end of the first year. So, its value at the end of the second year was:

Value at end of second year = $2100 + 80% of $2100
Value at end of second year = $2100 + $1680
Value at end of second year = $3780

Therefore, the actual percent gain or loss on the original $7000 investment is:

Actual percent gain or loss = ($3780 - $7000) / $7000 * 100%
Actual percent gain or loss = -46%

So, the actual percent loss on the original $7000 investment is 46%.

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The probability that a) the sample has a proportion between 0.5 and 0.7 is 0.780. b) The probability that the sample has a proportion within 5% is 0.819. c) The probability that the sample has a proportion less than 0.50 is 0.001. d) The probability that the sample has a proportion greater than 0.80 is 0.000.

a) To calculate this probability, we first need to standardize the interval (0.5, 0.7) using the formula: z = (p - P) / (σ / √(n))

where p is the sample b, P is the population proportion, σ is the standard deviation of sample proportions, and n is the sample size. Substituting the values, we get:

z1 = (0.5 - 0.62) / (0.8 / √(40)) = -2.24

z2 = (0.7 - 0.62) / (0.8 / √(40)) = 1.12

Using the standard normal table or calculator, the area between -2.24 and 1.12 is 0.780. Therefore, the probability that the sample has a proportion between 0.5 and 0.7 is 0.780.

b) The probability that the sample has a proportion within 5% of the population proportion is 0.819. We can find the range of sample proportions within 5% of the population proportion by adding and subtracting 5% of the population proportion from it, which gives: P ± 0.05P = 0.62 ± 0.031

The interval (0.589, 0.651) represents the range of sample proportions within 5% of the population proportion. To calculate the probability that the sample proportion falls within this interval, we standardize it using the formula above and find the area under the standard normal curve between -1.55 and 1.55, which is 0.819.

c) The probability that the sample has a proportion less than 0.50 is 0.001. To calculate this probability, we standardize the value of 0.50 using the formula above and find the area to the left of the resulting z-score, which is: z = (0.50 - 0.62) / (0.8 / √(40)) = -4.46

Using the standard normal table or calculator, the area to the left of -4.46 is 0.001. Therefore, the probability that the sample has a proportion less than 0.50 is 0.001.

d) The probability that the sample has a proportion greater than 0.80 is 0.000. To calculate this probability, we standardize the value of 0.80 using the formula above and find the area to the right of the resulting z-score, which is: z = (0.80 - 0.62) / (0.8 / √(40)) = 5.60

Using the standard normal table or calculator, the area to the right of 5.60 is very close to 0.000. Therefore, the probability that the sample has a proportion greater than 0.80 is 0.000.

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the backup goalie saved 42 shots. Answer: 42 shots. We can start by setting up an equation using the information given

what is equation ?

An equation is a mathematical statement that asserts that two expressions are equal. It is typically written with an equal sign (=) between the two expressions. For example, the equation 2x + 3 = 7 is a statement that asserts that the expression 2x + 3 is equal to 7.

In the given question,

We can start by setting up an equation using the information given:

save percentage = (number of shots saved / total number of shots attempted)

For the backup goalie, we know that their save percentage is 0.560, and we also know the total number of shots attempted on the goal is 75. Let's let the number of shots saved by the backup goalie be represented by the variable "t". Then we can write:

0.560 = t / 75

To solve for t, we can cross-multiply:

0.560 * 75 = t

t = 42

Therefore, the backup goalie saved 42 shots. Answer: 42 shots.

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If and only if the alternating sum of the two-digit chunks of a number n is divisible by m, then n is divisible by m.

Let's denote the two-digit chunks of n as a₁a₂, a₃a₄, ..., where a₁, a₂, a₃, a₄, ... are the digits from the ones place onward.

The alternating sum of these two-digit numbers is given by a₁a₂ - a₃a₄ + a₅a₆ - a₇a₈ + ...

We can rewrite n as (a₁a₂ × 100) + (a₃a₄ × 10) + (a₅a₆ × 1) + ...

The alternating sum expression can be written as (a₁a₂ × 100) - (a₃a₄ × 10) + (a₅a₆ × 1) - ...

If n is divisible by m, we have n ≡ 0 (mod m).

Rewriting n in terms of the alternating sum, we get (a₁a₂ × 100) - (a₃a₄ × 10) + (a₅a₆ × 1) - ... ≡ 0 (mod m).

Factoring out each two-digit chunk, we have (a₁a₂ - a₃a₄ + a₅a₆ - ...) ≡ 0 (mod m).

This implies that n is divisible by m if and only if the result of the alternating sum process is divisible by m.

m is the value that makes the alternating sum of the two-digit chunks of n a divisibility test for n.

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Step 1: Find the cost of 1 pie

1 pie = 4 cups of fruit + 1 pie crust

1 pie = 4(0.75) + 1(2.50)

1 pie = 3 + 2.50

1 pie = 5.50

Step 2: Find the amount of money a baker makes by selling 1 pie

1 pie cost = 5.50

1 pie revenue = 10.25

1 pie profit = 4.75

Step 3: Find the amount of money a baker makes by selling 10 pies

1 pie profit = 4.75

10 pies profit = 47.5

Answer: $47.50

Hope this helps!

Answer:

D. 4π

Step-by-step explanation:

Circumference: C = 2πr = 2π(8) = 16π

The distance = 1/4 circumference (angle is 90 degrees)

=> distance = 16π/4 = 4π

Answer:

[tex]\left(x\:+\:1\right)^2\:+\:y^2=25[/tex]

Step-by-step explanation:

The equation of a circle with radius r and center at (a, b) is given by
(x - a)² + (y - b)² = r²

Let's first find the radius

The circle intersects the x axis at two points (-6, 0) and (4, 0)

The diameter is therefore the absolute difference between the x values:
|-6 - 4| same as |4 - (-6)| = 10

The radius r = 5  (half of diameter)

Now, let's find the center point of the circle. This will lie midway between (-6, 0) and (4, 0)

Midpoint  (xm, ym) between two points(x1, y) and (x2, y2) :
xm = (x1 + x2)/2 = (-6 + 4)/2 = -1
ym = (y1 + y2)/2 = (0 + 0)/2 = 0

So the center (a, b) = (-1, 0) with a = -1, b = 0

The equation of the circle therefore is
(x - a)² + (y - b)² = r²
( x - (-1) )² + (y - 0)² = 25

(x + 1)² + y² = 25

To find a and b take any point (x, y) and plug these

Answer:

Step-by-step explanation:

In option (a), sin(t) < 0 and cos(t) < 0, In trigonometry, the terminal point of an angle t is the point on the unit circle where the angle intersects with the circle.

The position of the terminal point determines the quadrant in which the angle lies.

To determine the quadrant, we need to look at the signs of the sine and cosine functions. In quadrant I, both sine and cosine are positive. In quadrant II, sine is positive and cosine is negative. In quadrant III, both sine and cosine are negative. In quadrant IV, sine is negative and cosine is positive.

In option (a), sin(t) < 0 and cos(t) < 0, both the sine and cosine functions are negative. This means that the terminal point lies in quadrant III.

In option (b), sin(t) > 0 and cos(t) < 0, the sine function is positive and the cosine function is negative. This means that the terminal point lies in quadrant II.

In option (c), sin(t) > 0 and cos(t) > 0, both the sine and cosine functions are positive. This means that the terminal point lies in quadrant I.

In option (d), sin(t) < 0 and cos(t) > 0, the sine function is negative and the cosine function is positive. This means that the terminal point lies in quadrant IV.

In summary, the signs of the sine and cosine functions can be used to determine the quadrant in which the terminal point lies.

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Answer: 6cm

Step-by-step explanation: VF=(SF)^3

let x be the height of the larger container

VF1/VF2=(SF1/SF2)^3

54/128=(4.5/X)^3

rearrange to find x

x=cube root of (4.5^3*128)/54

x=6cm

If a number cube is rolled twice, then the probability of getting a six on the first role and then a number less than 5 on the second role is 1/9.

To find the probability of getting a six on the first roll and a number less than 5 on the second roll, we will multiply the individual probabilities of each event.

A number cube has 6 faces, so the probability of rolling a six is 1/6.

For the second roll, there are 4 numbers less than 5 (1, 2, 3, and 4), so the probability of rolling a number less than 5 is 4/6 or 2/3.

To find the combined probability, simply multiply the two probabilities: (1/6) × (2/3) = 2/18 = 1/9.

So, the probability of getting a six on the first roll and a number less than 5 on the second roll is 1/9.

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The probability that Julie got exactly seven questions correct is approximately 0.117 or 11.7%.

The probability of guessing the correct answer on any single true-false question is 1/2, and the probability of guessing the wrong answer is also 1/2.

The probability of guessing exactly seven questions correctly can be calculated using the binomial distribution formula:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

where:

n is the total number of questions, which is 10 in this case.

k is the number of questions guessed correctly, which is 7 in this case.

p is the probability of guessing any individual question correctly, which is 1/2 in this case.

Using the formula, we get:

P(X = 7) = (10 choose 7) * (1/2)^7 * (1/2)^(10-7)

= 120 * (1/2)^10

= 0.1171875

Therefore, the probability that Julie got exactly seven questions correct is approximately 0.117 or 11.7%.

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The coordinate of a point between A and B, such that the distance from A to point B is 3/11 of distance A to B, is 1.

Let's denote the unknown point between A and B as P, and let the distance from A to P be x. Then the distance from P to B is (11/3)x. Since the distance from A to B is 19 - (-3) = 22, we have the equation x + (11/3)x = 22(3/11), which simplifies to (14/3)x = 6, or x = 9/7. Therefore, the coordinate of point P is -3 + (9/7)(19 - (-3)) = 1.

To check our answer, we can verify that the distance from A to P is (10/7)(22) and the distance from P to B is (1/7)(22)(11), and that (10/7)(22) = (3/11)(22), which is indeed true.

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Answer:

  3.5 feet

Step-by-step explanation:

You want the difference in height between similar isosceles triangles, one with height of 30 ft and a base of 24 ft, the other with a side length of 28.5 ft.

We can find the side length of the larger triangle using the Pythagorean theorem. It will be ...

  longer side = √(30² +12²) ≈ 32.311 ft

Then the length x is the difference between the altitudes of the triangles. The altitudes are proportional to the side lengths, so we have ...

  (30 -x)/28.5 = 30/32.311

  x = 30-(28.5)(30/32.311) = 30(1 -28.5/32.311) ≈ 3.538 ≈ 3.5 . . . . feet

The hand rail is about 3.5 feet above the bridge deck.

We recognize that the distance from the hand rail to the top of the triangle is the product of the given side length (28.5 ft) and the cosine of the angle between the side and the altitude.

The tangent of that angle is the ratio of its opposite side (12 ft) to its adjacent side (30 ft), or θ = arctan(12/30).

The value of x is the difference of the altitudes of the triangles, so is ...

  x = 30 -28.5·cos(arctan(12/30)) ≈ 3.5 ft

We find this easier to compute.

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The sum of the interior angles of a triangle is 180 degrees.

Sure, here's a question related to the Triangle Sum and Exterior Angle Theorem: Consider triangle ABC. The measure of angle A is 60 degrees, and the measure of angle B is 80 degrees. What is the measure of angle C? Using the Triangle Sum Theorem, we know that the sum of the interior angles of a triangle is always 180 degrees.

Additionally, the Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.  

Based on this information, determine the measure of angle C in triangle ABC and provide a step-by-step explanation of how you arrived at your answer.

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it is 16 steps from second base to third base .we can get this answer by

solving the logic given in the question .we need to add up the number of steps

what is  add up  ?

"Add up" means to calculate the total of two or more numbers or quantities by combining them together. It involves the mathematical operation of addition, which is the process of finding the sum of two or more numbers. For example, if you add up the numbers 3, 5, and 7, t

In the given question,

To find out how many steps it is from second base to third base, we need to add up the number of steps Gavin took in each direction.

Starting at second base, he took 18 steps forward to get caught, and then took 7 steps back. This leaves him 11 steps forward from second base.

Next, he took 3 more steps forward, for a total of 14 steps forward from second base. But then he took 5 steps back, leaving him 9 steps forward from second base.

Then he took 11 more steps forward, for a total of 20 steps forward from second base. But then he took 4 steps back, leaving him 16 steps forward from second base.

Finally, he was tagged out halfway between second and third base. Since he was 16 steps forward from second base, and halfway between second and third base, we can assume that third base is 16 steps away from second base.

Therefore, it is 16 steps from second base to third base.

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1. The area of the shaded region is

2. percentage of the shaded part is 89.6%

The area of a shape is the space occupied by the boundary of a plane figures like circles, rectangles, and triangles.

The area of the shaded part = area of the rectangle - area of unshaded part

Area of rectangle = 7× 11 = 77 unit²

area of rectangle = 1/2 bh

= 1/2 × 4 × 4

= 1/2 × 8

= 4 unit²

area of second triangle = 4 units²

area of unshaded part = 4+4 = 8 units²

area of shaded part = 77-8 = 69units²

2. percentage of the rectangle shaded = 69/77 × 100

= 6900/77 = 89.6%

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The number of cells found in an average Koala is 1.30 x 10¹², under the condition that a cell is a sphere with radius 10 or 0. 001 centimeter.

Then the volume of a sphere with radius 10 cm is considered to be 4/3π(10)³ cubic cm that is approximately 4,188.79 cubic cm.
The evaluated volume of a sphere with radius 0.001 cm is 4/3π(0.001)³ cubic cm that is approximately 0.00000419 cubic cm.

Then the evaluated mass of a single cell is  found by applying the formula
mass = density x volume
In case of larger cell, the mass will be
mass = 1.1 g/cm³ x 4,188.79 cubic cm
= 4,607.67 grams

In case of  smaller cell, the mass will be

mass = 1.1 g/cm³ x 0.00000419 cubic cm
= 0.00000461 grams

As koalas measure an average of 6 kilograms or 6,000 grams², we can evaluate the number of cells in an average koala using division of the weight of the koala by the mass of a single cell

In case of larger cells
number of cells = weight of koala / mass of single cell
number of cells = 6,000 grams / 4,607.67 grams
≈ 1.30 x 10⁶ cells

For smaller cells:
number of cells = weight of koala / mass of single cell
number of cells = 6,000 grams / 0.00000461 grams

≈ 1.30 x 10¹² cells
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⁶

There are 4,096,000,000 possible passwords of length 6 using special characters, digits, and letters, with characters allowed to be repeated.

To compute the number of passwords possible with a length of 6 using digits, letters, and special characters, with characters allowed to be repeated, follow these steps:

1. Count the number of options for each character type:
  - Digits: 10 (0-9)
  - Letters: 26 (a-z)
  - Special characters: 4 (*, &, $, #)

2. Combine the options for all character types:
  Total options per character = 10 digits + 26 letters + 4 special characters = 40

3. Calculate the number of possible passwords:
  Since characters may be repeated and the password has a length of 6, the number of possible passwords = 40^6 (40 options for each of the 6 character positions)

4. Calculate the result:
  Number of possible passwords = 40^6 = 4,096,000,000

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The value of integral of (198r² + 9r – 50 dr  is  (16/27)ln|11r + 2| - (2/3)ln|9r - 4| + C

Now, we can write the integrand as a sum of two fractions:

(198r² + 9r - 50) / (11r + 2)(9r - 4) = A/(11r + 2) + B/(9r - 4)

To find A and B, we need to solve for them using the method of equating coefficients:

198r² + 9r - 50 = A(9r - 4) + B(11r + 2)

Setting r = 4/9 and r = -2/11, we get two equations:

198(4/9)² + 9(4/9) - 50 = A(9(4/9) - 4) + B(11(4/9) + 2)

198(-2/11)² + 9(-2/11) - 50 = A(9(-2/11) - 4) + B(11(-2/11) + 2)

Solving these equations gives A = 16/27 and B = -2/3.

So, the integral can be written as:

∫(198r² + 9r - 50) / (11r + 2)(9r - 4) dr = ∫16/27(11r + 2)^-1 dr - ∫2/3(9r - 4)^-1 dr

Integrating each term gives:

(16/27)ln|11r + 2| - (2/3)ln|9r - 4| + C

where C is the constant of integration.

In summary, using partial fraction decomposition, we can express the given integral as a sum of two simpler integrals, which can be evaluated using the natural logarithm function.

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An inequality that can be used to determine x, the number of hours before Marcus will need to add chemicals to maintain the pH for the pool would be 6.9 + 0.05x ≤ 7.8

To determine the number of hours (x) before Marcus will need to add chemicals to maintain the pool's pH, we can use an inequality with the given information.

-The maximum pH value allowed for the pool is 7.8.
-The pool currently has a pH value of 6.9.
-The pH value of the pool increases by 0.05 per hour.

The inequality for this scenario would be:

6.9 + 0.05x ≤ 7.8

This inequality states that the current pH value (6.9) plus the increase in pH per hour (0.05x) should be less than or equal to the maximum allowed pH value (7.8). This will help us determine the number of hours (x) before Marcus needs to add chemicals to maintain the pH for the pool.

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The equation of the tangent line to the curve at P(2, -12) is y = 2x - 16.

(a) To find the slope of the curve y = x^3 - 10x at the given point P(2, -12), we need to find the derivative of the function y with respect to x, and then evaluate it at x = 2.

Step 1: Find the derivative, dy/dx
y = x^3 - 10x
dy/dx = 3x^2 - 10

Step 2: Evaluate the derivative at x = 2
dy/dx (2) = 3(2)^2 - 10 = 12 - 10 = 2

The slope of the curve at P(2, -12) is 2.

(b) To find an equation of the tangent line to the curve at P(2, -12), we'll use the point-slope form of the equation: y - y1 = m(x - x1).

Step 1: Use the slope found in part (a) and the given point P(2, -12).
m = 2
x1 = 2
y1 = -12

Step 2: Plug the values into the point-slope equation.
y - (-12) = 2(x - 2)
y + 12 = 2x - 4

Step 3: Rearrange the equation to get the final form.
y = 2x - 4 - 12
y = 2x - 16

The equation of the tangent line to the curve at P(2, -12) is y = 2x - 16.

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