In a certain class of 40students, 90% passed ssce mathematics examinations and 75% passed English. If 2 students failed both mathematics and English, what percentage of students passed both examinations
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It will take approximately 8 minutes to fill the water truck, and the weight of the water load will be approximately 30,296 pounds.

First, we need to convert the dimensions of the tank from inches to feet:

68 inches diameter = 68/12 feet = 5.67 feet diameter

24 feet long = 24 feet long

Next, we can calculate the volume of the tank in cubic feet:

Since 1 cubic foot of water weighs 62.4 pounds, we can calculate the weight of the water in the tank in pounds:

To find out how long it will take to fill the tank, we can use the flow rate of the fire hydrant:

Finally, we can divide the volume of the tank by the flow rate to find out how long it will take to fill the tank:

Therefore, it will take approximately 8 minutes to fill the water truck, and the weight of the water load will be approximately 30,296 pounds.

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Answer:

Kids's Kingdom ordered 170 large stuffed animals and 280 small stuffed animals. The dollar amount of each size ordered was $3,400 for the large stuffed animals and $3,920 for the small stuffed animals.

Step-by-step explanation:

Let's use the following variables:

L for the number of large stuffed animals

S for the number of small stuffed animals

We can set up a system of two equations to represent the given information:

L + S = 450 (equation 1)

20L + 14S = 7320 (equation 2)

We can solve this system of equations using substitution or elimination. Let's use substitution.

From equation 1, we can solve for L:

L = 450 - S

Substitute this expression for L into equation 2:

20(450 - S) + 14S = 7320

Distribute the 20:

9000 - 20S + 14S = 7320

Simplify and solve for S:

6S = 1680

S = 280

So, Kids's Kingdom ordered 280 small stuffed animals. We can use equation 1 to find the number of large stuffed animals:

L + 280 = 450

L = 170

Therefore, Kids's Kingdom ordered 170 large stuffed animals.

To find the dollar amount of each size ordered, we can multiply the number of each size by the cost per item:

170 large stuffed animals at $20 each: 170 * $20 = $3,400

280 small stuffed animals at $14 each: 280 * $14 = $3,920

So, Kids's Kingdom spent $3,400 on large stuffed animals and $3,920 on small stuffed animals for a total cost of $7,320.

Using Hooke's law the maximum value of |cos(8t)| is 1, so the maximum excursion is 1/6 meters.

To find the spring constant k, we use Hooke's law:

F = -ky

where F is the weight of the object, and y is the distance it is stretched from its rest position. At equilibrium, F = mg = 10 × 9.81 = 98.1 N. Thus,

98.1 = -k × 0.098

k = -1000 N/m

The equation of motion for the system is given by:

my'' + ky = F(t)

Substituting the given values, we get:

10y'' + (-1000)y = 100cos(8t)

y'' - 100y = 10cos(8t)

with initial conditions y(0) = 0 and y'(0) = 0.

The characteristic equation is r² - 100 = 0, with roots r = ±10i. The complementary solution is therefore y_c(t) = c1cos(10t) + c2sin(10t).

For the particular solution, we assume a form of yp(t) = Acos(8t) + Bsin(8t), and substitute it in the differential equation to get:

-64Acos(8t) - 64Bsin(8t) - 100(Acos(8t) + Bsin(8t)) = 10cos(8t)

Solving for A and B, we get A = -1/6 and B = 0. Thus, the particular solution is yp(t) = (-1/6) × cos(8t).

The general solution is therefore y(t) = c1cos(10t) + c2sin(10t) - (1/6)*cos(8t). Applying the initial conditions, we get c1 = 0 and c2 = 0, so the particular solution is simply y(t) = (-1/6) × cos(8t).

The maximum excursion from equilibrium can be found by taking the absolute value of y(t) and finding its maximum value. We have:

|y(t)| = (1/6) × |cos(8t)|

The maximum value of |cos(8t)| is 1, so the maximum excursion is 1/6 meters.

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Answer: Total number of males= C

Step-by-step explanation:

The length of the arc is approximately 5.51 meters when a circle with a radius of 21 meters is subtended by a central angle of 15 degrees.

The length of an arc of a circle with radius 21m that subtends a central angle of 15° can be calculated using the formula:

Arc length = (central angle/360°) x 2πr

where r is the radius of the circle, and π is the mathematical constant pi.

Substituting the given values, we get:

Arc length = (15/360) x 2π x 21

Arc length = (1/24) x 2 x 3.14 x 21

Arc length = (1/12) x 3.14 x 21

Arc length = 5.51 meters (rounded to two decimal places).

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We need to apply the  formula to determine the length of a circle's arc: (Central angle / 360°) x (2 x x radius) is the formula for arc length. the radius is  distance from the circle center to any point on its perimeter,

and the central angle is the angle subtended by the arc at its center. The radius in this instance is stated as 21 meters, while the arc's center angle is provided as 15 degrees. When these values are added to the formula, we obtain: arc length is equal to (15°/360°) x (2x x 21m) 3.68 m. As a result, the arc measures around 3.68 meters in length. As a result, the radius is the distance from the circle's center to any point on its perimeter, if we were to sketch an arc of The arc's length would be around 3.68 meters for a circle with a radius of 21 meters and a center angle of 15 degrees.

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The value of y in the given triangle is 7.44 units.

The trigonometric ratio between the opposing and adjacent sides of a right triangle that contains an angle is its tangent. The trigonometric functions, also known as circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths. All geosciences, including navigation, solid mechanics, celestial mechanics, geodesy, and many more, utilise them extensively.

The given triangle is a right-triangle.

The trigonometric identity that gives the relationship between opposite side and adjacent side is tan.

Thus,

tan (28) = opposite / adjacent = y / 14

y = 0.53 (14)

y = 7.44

Hence, the value of y in the given triangle is 7.44 units.

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Answer:

2)She could either divide by 3 or distribute 3 into the parenthesis

Answer:

x = 24 and y = 24

Step-by-step explanation:

Let's use algebra to solve this optimization problem.

Let x be the first number, and y be the second number. Then we have the following two equations based on the problem statement:

x + y = 48 (sum of the two numbers is 48)

xy = ? (product of the two numbers, which we want to maximize)

To solve for x and y in terms of each other, we can use the fact that:

(x + y)^2 = x^2 + 2xy + y^2

Expanding the left side of the equation gives:

x^2 + 2xy + y^2 = 2304

And substituting xy for its value in terms of x and y gives:

x^2 + 2xy + y^2 = x^2 + 2(48 - x)y + y^2 = 2304

Simplifying this equation gives:

2y^2 - 96y + x^2 - 2304 = 0

To maximize the product xy, we need to maximize the value of xy = x(48 - x) = 48x - x^2. This function is a quadratic that opens downwards, and therefore, its maximum value occurs at the vertex of the parabola, which is located at x = -b/2a = -48/(2*-1) = 24.

Thus, the two positive real numbers that sum up to 48 and their product is a maximum are x = 24 and y = 24.

The area of the parallelogram is 324 square yards.

What is a parallelogram ?

A parallelogram is a four-sided geometric shape that has two pairs of parallel sides. It isdefined by its four vertices, four sides, and two diagonals that intersect at their midpoint. The opposite sides of a parallelogram are congruent, and the opposite angles are also congruent. The adjacent angles are supplementary and add up to 180 degrees.

To find the area of a parallelogram, we need to multiply the length of the base by the height, which is the perpendicular distance from the base to the opposite side. This formula is similar to finding the area of a rectangle, where the base is one side of the rectangle and the height is the distance from that side to the opposite side.

Calculating the area of the given parallelogram :

The area of a parallelogram is  A = bh, where b is the base and h is the height.

Given the height of the parallelogram is 12 yards and the base is 27 yards. Using the formula for the area of a parallelogram, we can calculate the area as follows:

A = bh

A = 27 yards × 12 yards

A = 324 square yards

Therefore, the area of the parallelogram is 324 square yards.

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3. The statement is true,  the correlation coefficient is close to -1. 4. temperature for a city with a latitude of 48 is 43. 5. The statement is false. 6. cannot make a reasonable estimate

Equations can be used to model real-world situations and solve problems in many fields, including science, engineering, finance, and more. They are an essential tool in mathematics and are used extensively in algebra, calculus, and other advanced branches of math.

Equations can involve various mathematical operations such as addition, subtraction, multiplication, division, exponentiation, and others.

Question 3:

The statement is true. We can check this by calculating the correlation coefficient between the latitude and temperature data points, which should be close to -1. The calculated line of best fit is also consistent with the given data.

Question 4:

To estimate the temperature for a city with a latitude of 48, we can use the equation of the line of best fit:

y = -1.07x + 92.87

Substituting x = 48, we get:

y = -1.07(48) + 92.87

y = 42.79

Rounding to the nearest whole number, the estimated temperature for a city with a latitude of 48 is 43.

Question 5:

The statement is false. We can check this by calculating the correlation coefficient between the passengers and suitcases data points, which should be close to 1. The given line of best fit has a negative slope, which is inconsistent with the positive correlation between the variables.

Question 6:

To estimate the number of suitcases for a flight carrying 250 people, we can use the equation of the line of best fit:

y = -1.98x + 7.97

Substituting x = 250, we get:

y = -1.98(250) + 7.97

y = -485.03

However, it does not make sense for the number of suitcases to be negative. Therefore, we cannot make a reasonable estimate for the number of suitcases on a flight carrying 250 people using this line of best fit.

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Answer:

82

Step-by-step explanation:

divide this shape into a triangle and rectangle and use 24 as the base of the triangle using the Pythagoras theorem you get that the perpendicular is 7 and since 13 is also parallel to that side, 13+7=20 so one side equals 20 one side equals 25 one side equals 24 and one side equals 13 now add all of these to get 82

Step-by-step explanation:

See image below ..... perimeter = 25 + 13 + 24 + 13 + x

  x is found using pythagorean theorem for right triangles

x = 7 units

total perimeter is then 82 units

Answer:

Troy scored 277 points in the third game of pinball.

Step-by-step explanation:

Let x be the number of points Troy scored in the third game.

We know that Troy scored a total of 945 points in 3 games, so we can set up an equation:

312 + 356 + x = 945

Simplifying this equation, we get:

668 + x = 945

Subtracting 668 from both sides, we get:

x = 277

Therefore, Troy scored 277 points in the third game of pinball.

The equation for the ellipse is: [tex]$\frac{x^2}{37}[/tex] + [tex]$\frac{y^2}{25}[/tex] [tex]= 1$$[/tex]

The standard equation for an ellipse centered at the origin is:

[tex]$\frac{x^2}{a^2}[/tex] + [tex]$\frac{y^2}{b^2}[/tex] [tex]= 1$$[/tex]

where a is the distance from the center to a vertex, and b is the distance from the center to a co-vertex.

In this case, the vertices are located at [tex]$(\pm\sqrt{37}, 0)$[/tex], which means [tex]$a=\sqrt{37}$[/tex]. The distance between the foci is [tex]$2c=2\sqrt{12}=2\sqrt{3\times 4}=2\sqrt{3}\times 2=4\sqrt{3}$[/tex], which means [tex]$c=2\sqrt{3}$[/tex].

The value of b can be found using the relationship between a, b, and c in an ellipse:

[tex]$$a^2 = b^2 + c^2$$[/tex]

Substituting the values we know, we get:

[tex]$$37 = b^2 + (2\sqrt{3})^2$$[/tex]

Simplifying:

[tex]$$37 = b^2 + 12$$[/tex]

[tex]$$b^2 = 37 - 12 = 25$$[/tex]

Taking the square root of both sides, we get:

[tex]$$b = \pm 5$$[/tex]

Since the co-vertices are located at [tex]$(0,\pm b)$[/tex], we can see that [tex]$b=5$[/tex] (and not -5, since the ellipse is centered at the origin).

Therefore, the equation for the ellipse is:

[tex]$\frac{x^2}{37}[/tex] + [tex]$\frac{y^2}{25}[/tex] [tex]$ = 1$$[/tex]

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Answer:

The pink triangle

Step-by-step explanation:

IMPORTANT NOTE: Make sure all the units are the same and consistent

Perimeter of a figure = Total length of the outer boundary

Shape of each figure in this question = Isosceles Triangle

Perimeter of triangle = Sum of all three sides

Perimeter of pink triangle = 36m + 36m + 20m

                                             = 92m

Perimeter of green triangle = 25m + 25m + 35m

                                             = 85m

∴Comparing the two values calculated above, it can be observed that the pink triangle has a greater perimeter

Grace need to buy the jump rope, the game, and the ball $8.

$7 will get you a jump rope.

$10 will get you a board game.

$4 will get you a playground ball.

total amount to be spent: $7 + $10 + $4 = $21

She has ten dollars.

$3 was all her uncle gave her.

13 dollars are all she has.

She needed $8, therefore 21 - 13 = $8.

a sum of money awarded as compensation, a bounty, or to cover costs. a wage that comes with a cost-of-living supplement. especially: a regular amount set aside for household or personal costs.

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The degree measure of the largest interior angle of the given heptagon is 132.57 degrees.

The largest interior angle of the given heptagon. The degree measures of the interior angles of a heptagon are 7, with 7 sides or vertices, are $x, ~x, ~x-2, ~x-2, ~x+2, ~x+2,$ and $x+4$ degrees.

The sum of the degree measures of the interior angles of a polygon with n sides is given by $S = (n-2) \c dot 180^\circ$. The sum of the interior angles of a heptagon is given by $S = (7-2) \cdot 180^\circ = 900^\circ$.

The sum of the degree measures of the interior angles of the heptagon is equal to $x+x+x-2+x-2+x+2+x+2+x+4 = 7x+4$. To find the value of x, we will set this equation equal to the total sum of the interior angles:$7x+4 = 900^\circ$. Solving for x, we get$x = 128.57$

We may now substitute the value of x to get the degree measures of each of the angles in the heptagon:$x = 128.57^\circ$$x = 128.57^\circ$$x - 2 = 126.57^\circ$$x - 2 = 126.57^\circ$$x + 2 = 130.57^\circ$$x + 2 = 130.57^\circ$$x + 4 = 132.57^\circ$

To find the degree measure of the largest interior angle, we must look for the angle with the largest value. We can see that the largest angle measures $132.57^\circ$.

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In both cases, the slope (m) is the coefficient of the x-variable, and the y-intercept (b) is the constant. In the first equation, the slope (m) is 2, and the y-intercept (b) is -3. In the second equation, the slope (m) is 3, and the y-intercept (b) is 6.

Graph is a data structure that consists of nodes (vertices) and edges (connections between nodes). Graphs are used to represent networks, such as social networks, transportation networks, and communication networks. Graphs provide a way to visualize relationships between data points and can be used to discover patterns in data and make predictions. Graphs are a powerful tool in data science and can be used to analyze complex systems.

This is happening because the equations represent the linear relationships between the variables of the graphs. The first equation is in slope-intercept form, and can be used to calculate the y-value of a given x-value. The equation for the second graph is also in slope-intercept form, and can be used to calculate the y-value of a given x-value.

In both cases, the slope (m) is the coefficient of the x-variable, and the y-intercept (b) is the constant. In the first equation, the slope (m) is 2, and the y-intercept (b) is -3. In the second equation, the slope (m) is 3, and the y-intercept (b) is 6.

These equations represent the linear relationships between the variables of the graphs. The slope of the line is the rate of change between the x- and y-values, while the y-intercept is the starting point of the line. By using the equations, we can calculate the y-value of any given x-value on the graph, and thus accurately represent the linear relationships between the variables.

The diagram is given below.

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Complete questions as follows-

Answer:

4 apples

Step-by-step explanation:

We know

John had 3 apples, then Dropped 2, found 4, then gave 1 to his friend.

How many apples does he have now​?

3 - 2 + 4 - 1 = 4 apples

So, he has 4 apples now.

Answer:

4 apples

Step-by-step explanation:

We know that John had 3 apples

but then he had dropped 2 of them

he then found 4

then he gave 1 to his friend

3-2=1

1+4=5

5-1=4

The answer is 4

Hope this helps!

The value of k in the cubic equation is 2 and factorization of the equation is 2(x-1)^2(x+1)

If 2x^3 + 15x^2 + 24x + k has three linear factors, then it must be in the form:

[tex]2x^3 + 15x^2 + 24x + k = a(x-b)(x-c)^2[/tex]

where a, b, and c are constants, and (x-c)^2 means that (x-c) is a repeated factor.

Expanding the right side, we get:

[tex]2x^3 + 15x^2 + 24x + k = a(x^3 - 2cx^2 + cx^2 - 2cx + c^2x - c^2)[/tex]

Simplifying, we get:

[tex]2x^3 + 15x^2 + 24x + k = ax^3 - 2acx^2 + ac^2x - 2acx + ac^2 + a(-c^2)\\2x^3 + 15x^2 + 24x + k = ax^3 + (-2ac + ac^2)x^2 + (-2ac + ac^2)x - ac^2[/tex]

Equating the coefficients of each power of x on both sides, we get:

[tex]a = 2\\-2ac + ac^2 = 15\\-2ac + ac^2 = 24\\-ac^2 = k\\\\[/tex]

From the second and third equations, we can see that -2ac + ac^2 = 15 and -2ac + ac^2 = 24. These equations are equivalent, so we can set them equal to each other:

[tex]-2ac + ac^2 = 15 = 24[/tex]

Simplifying, we get:

ac^2 - 2ac + 9 = 0

This is a quadratic equation in ac. Solving for ac using the quadratic formula, we get:

[tex]ac = [2 \± \sqrt(4 - 4(1)(9))] / 2[/tex]

[tex]ac = 1 \± \sqrt(-5)[/tex]

Since ac is the product of two real numbers (b and c), we know that the square root of -5 must cancel out. This is only possible if ac = 1 and c = -1. Therefore, b = -c = 1, and we can find k by setting ac^2 = -k:

[tex]1 = 2a\\15 = -2ac + ac^2 = -2a + a^2\\24 = -2ac + ac^2 = -2a + a^2\\-k = -ac^2 = 2(-1)^2 = 2\\[/tex]

Therefore, k = -2, and the factorization of 2x^3 + 15x^2 + 24x - 2 is:

2x^3 + 15x^2 + 24x - 2 = 2(x-1)^2(x+1)

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The method we would be using here to prove the triangle's congruency is LA, the leg-acute theorem.

The dimensions of the sides and angles of two or more triangles determine whether they are congruent or not. A triangle's size and shape are determined by its three sides and three angles, respectively. If pairings of corresponding sides and corresponding angles are equal, two triangles are said to be congruent. They are the exact same size and form.

Here in the given figures, we can use the LA theorem also known as leg acute. It says that a right triangle is congruent if its leg and any acute angle are congruent with the corresponding leg and any acute angle of another right triangle.

The hypotenuse of the first triangle is corresponding to the hypotenuse of the second triangle.

The angle opposite to the right angle in both the triangles are corresponding to each other.

So as per the LA theorem we can prove that the triangle is congruent.

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The solutions to the system equations are x = -1 and x = 0.

When two or more variables are related to one another and equations are constructed to determine each variable's value, the result is a system of equations. an equation is a balance scale, and for the equation to stay true, both sides must be equal.

Given equations are:

[tex]y=2^x - 1\\y=\frac{1}{2} x[/tex]

The solutions of the system for which; [tex]y_{1} =y_{2}[/tex]

According to the given table, the solutions to the system of both equations at x = -1 and x = 0.

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An 1 οr perhaps an even integer will be drawn back 75% οf the times frοm the set οf 1, 2, 3 and 4. Thus, Prοbability P(nοt even) is 75%.

Prοbability is the likelihοοd that sοmething will οccur οr the prοbability that sοmething will happen. Prοbability is the measure οf hοw prοbable it is that a cοin will land heads up after being tοssed intο the air.

As prοbabilistic arguments sοmetimes prοduce οutcοmes that appear cοntradictοry οr illοgical, prοbability is usually regarded as amοng the mοst challenging tοpics οf mathematics.

P(1) = 1/4(there is one card with a 1)

P(even) = 2/4 = 1/2 (there are 2 cards with even numbers out of 4)

Therefore,

P( 1 or even) =P(1) + P(even)

= 1/4 + 1/2

= 3/4

To express the answer as a percentage, we can multiply by 100:

P(1 or even) = 3/4 × 100%

=> 75%

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You pick a card at random. card 1, card 2 and card 3

What is P(not even)?

Write your answer as a fraction or percentage.

The monthly profit, P(x), of a pizza parlor is given by the function P(x) = 5.75x - 1,200, where x is the number of pizzas sold, given revenue and expense functions.

The monthly profit, P(x), is the difference between the revenue and the expenses, so:

P(x) = R(x) - E(x)

From the given information:

R(x) = 12.5x (revenue per pizza is $12.5, and x is the number of pizzas sold)

E(x) = 1,200 + 6.75x (expenses, which include the monthly rent of $1,200 and the production cost of $6.75 per pizza, multiplied by the number of pizzas sold)

Substituting the values of R(x) and E(x) into the equation for P(x):

P(x) = 12.5x - (1,200 + 6.75x)

Simplifying the equation:

P(x) = 5.75x - 1,200

Therefore, the function representing the monthly profit, P(x), is P(x) = 5.75x - 1,200.

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Answer:

P(x)= 5.75x -1,200

Step-by-step explanation:

Plato/Edmentum

Parts-R-Us could be eligible for a $0 bonus if it achieves an on-time performance rate of 94% or more.

The company would have to maintain an order delivery rate of 95% in order to qualify for the bonus. If achieved, the bonus would be as follows: Bonus = 94% * (1 - 100).

The following is the calculation for the bonus:

Bonus = P * (1 - D)

Where:

P represents the proportion of orders that are delivered on time.

D represents the total number of orders placed late.

In the event that Parts-R-Us achieves an on-time performance rate of 94%, the bonus would be as follows:

Bonus = 94% * (1 - 100)

Bonus = $0

If the company only managed to complete its tasks 94% of the time, it will not be eligible for a bonus.

In order to qualify for the maximum bonus, Parts-R-Us would have to maintain an on-time delivery rate of 95%.

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In the given system of equations one taco costs $0.80 and one burrito costs $0.72.

An equation system is a finite collection of equations for which we searched for the common solutions. It is sometimes referred to as a set of simultaneous equations or an equation set. The classification of a system of equations is similar to that of a single equation. In modelling issues where the unknown values may be expressed in the form of variables, a system of equations finds use in everyday life.

Let us suppose the cost of one taco = x.

Let us suppose the cost of one burrito = y.

Then, for Emma we have:

3x + y = 3.65

For Cooper we have:

x + 2y = 3.30

Using elimination, multiply the first equation by 2 and subtract it from the second equation:

(2)(3x + y = 3.65)

6x + 2y = 7.30

x + 2y = 3.30

-5x = -4

x = 4/5

Substituting this value of x into either equation:

3(4/5) + y = 3.65

y = 2.15/3 ≈ 0.72

Therefore, one taco costs $0.80 and one burrito costs $0.72.

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[tex]\frac{1}{4} n - 7 \leq 95[/tex]

After the time of seven hours, the cream pie would have approximately 768 S. aureus cells after 7 hours with a generation time of 60 minutes.

Six S. aureus cells have been accidentally inoculated into a cream pie. S. aureus has a generation time of 60 minutes. S. aureus is a pathogenic bacterium found in the environment, as well as on the skin, and in the upper respiratory tract.

The generation time of this bacterium is 60 minutes, meaning that a single bacterium can produce two new cells in 60 minutes.

If there are 6 S. aureus cells in a cream pie, the number of bacteria will continue to increase as time passes.

The number of generations (n) in seven hours is calculated as:

n = t/g

n = 7 hours × 60 minutes/hour/60 minutes/generation = 7 generations

The number of cells in the cream pie after 7 hours is calculated as :

N = N₀ × 2ⁿ

N = 6 cells × 2⁷

N = 768 cells

Therefore, after seven hours, the cream pie would have approximately 768 S. aureus cells.

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Answer:

16 feet

Step-by-step explanation:

Area of triangle = 1/2(a x b)

A = 1/2(6.4 x 5)

A = 32/2

A = 16 feet

The result (recurrent value), A = sum j=1 to 89 ln(j), is true for every n. This is the desired result.

As in T(n) = T(n/2) + n, T(0) = T(1) = 1, a recurrence or recurrence relation specifies an infinite sequence by explaining how to calculate the nth element of the sequence given the values of smaller members.

We can start by proving the base case in order to demonstrate the first portion through recurrence. Let n = 1. Next, we have:

Being true, ln(a1) = ln(a1). If n = k, let's suppose the formula is accurate:

Sum j=1 to k ln = ln(prod j=1 to k aj) (aj)

Prod j=1 to k aj * ak+1 = ln(prod j=1 to k+1 aj)

(Using the logarithmic scale) = ln(prod j=1 to k aj) + ln(ak+1)

Using the inductive hypothesis, the property ln(ab) = ln(a) + ln(b)) = sum j=1 to k ln(aj) + ln(ak+1) = sum j=1 to k+1 ln (aj)

(b), we can use the just-proven formula:

A = ln(1, 2,...) + ln + ln (89)

= ln(j=1 to 89) prod

sum j=1 to 89 ln = ln(prod j=1 to 89 j) (j).

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