Given that, 2023 can be written as a1 + a2 + a3 + · · · + an where n ≥ 2 is a positive integer, a1 ≤ a2 ≤ a3 ≤ · · · ≤ an are positive integers, and a1 ≥ an−1. We need to determine the number of ways in which 2023 can be written in such a manner.
Let's apply the formula for finding the number of ways to represent an integer, where each term in the sum is an increasing sequence of positive integers.
In general, the formula for the number of ways to represent n is the number of partitions of n into an increasing sequence of positive integers. Let p(n) denote the number of partitions of n into an increasing sequence of positive integers.
Then, the number of ways to represent n as an increasing sequence of positive integers is given by p(n) - 1, as we cannot use the representation where n is the only term in the sum.
If we find p(2023), we can find the number of ways to represent 2023 as an increasing sequence of positive integers. Therefore, p(2023) - 1 is the required number of ways to represent 2023.
Let's calculate p(2023). The easiest way to calculate p(2023) is by generating functions. Since we are looking for an increasing sequence, we can use the formula for the generating function for partitions into distinct parts, which is:
(∑n=0∞xn)/(1−x)=1+x+x2+x3+x4+⋯.
We can replace x^n in the numerator with a generating function for the sum of the partitions of n into increasing parts to get the desired generating function.
(∑n=0∞x(n2+n)/2)/(1−x)=1+x+2x2+3x3+4x4+⋯.
The numerator in the generating function is (∑n=0∞x(n2+n)/2)=(∑n=0∞x(n2/2+n/2))=(∑n=0∞x(n/2)2+(1/4)(n+1/2))=(∑n=0∞(x1/4)n(n+1)+(x1/4)n+1/2)/2. The numerator is a sum of two geometric series, so we can simplify it.
(∑n=0∞(x1/4)n(n+1)+(x1/4)n+1/2)/2=(1/(1−x/4))^2+(x/2(1−x/4)^2).
(x/2(1−x/4)^2)=x(1/(1−x/4))^2−(x/2)/(1−x/4).
Therefore,
(∑n=0∞xn(n+1)/2)/(1−x)=p(0)+p(1)x+p(2)x2+p(3)x3+⋯=((1/(1−x/4))^2)−(x/2(1−x/4)^2).
The coefficient of x^2023 is x^2023−2−x/2(1−x/4)^2. Since x^2023−2=691104804/4^2, x^2022^2=−2022/4, and x^2021^2=303805/16, the required number of ways to represent 2023
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The angle measures of $\triangle ABC$ are $A=54\degree$ , $B=38\degree$ , and $C=88\degree$. List the sides of the triangle in order from shortest to longest
The triangle's sides are b, a, and c, listed from shortest to longest.
The side of triangle ABC opposite angle A is indicated by the letter a.
The letter b stands for the side opposite angle B.
The letter c stands for the side that faces angle C.
The triangle inequality theorem, states that, in order to establish the order of the sides from smallest to longest,
The largest side is the one opposite the largest angle and vice versa.
Hence, we have
B = 38 degrees, thus b.
a = 54 degrees from A
c from C equals 88 degrees.
As a result, the sides are arranged in the following order: b a c.
In other words, the side across from angle B is the smallest, followed by the side across from angle A, and the side across from angle C is the longest.
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I NEED HELP ON THIS ASAP!
A graph of each inequality is shown in the grid below.
A shape which the solution region represent is a rectangle.
How to determine and graph the solution for this system of inequalities?In order to graph the solution for the given system of inequalities on a coordinate plane, we would use an online graphing calculator to plot the given system of inequalities and then take note of the point of intersection;
0.4r ≤ 120 .....equation 1.
r ≥ 4(360/5) .....equation 2.
By evaluating the given system of inequalities, we have;
r ≤ 120/0.4
r ≤ 300
r ≥ 4(360/5)
r ≥ 4(72)
r ≥ 288
By combining the system of inequalities, we have:
288 ≤ r ≤ 300
Based on the graph (see attachment), we know that the solution to the given system of inequalities is the shaded region between the solid line, as well as the point of intersection of the lines representing each inequality.
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Calculate the area of triangle of PQR correct to 3 significant figure if P=8. 5 cm and Q=6. 8 and R=68. 4°
The area of triangle PQR, rounded to 3 significant figures, is approximately 29.6 square centimeters.
How to find the area of triangle PQR?To find the area of triangle PQR, we can use the formula:
Area = 1/2 * PQ * QR * sin(R)
where PQ and QR are the lengths of sides PQ and QR, and R is the angle between them in degrees.
Substituting the given values, we get:
Area = 1/2 * 8.5 cm * 6.8 cm * sin(68.4°)
Using a calculator, we can find that:
sin(68.4°) ≈ 0.933
So, the area of the triangle is:
Area ≈ 1/2 * 8.5 cm * 6.8 cm * 0.933
Area ≈ 29.63 cm²
Rounding this to 3 significant figures, we get:
Area ≈ 29.6 cm²
Therefore, the area of triangle PQR, rounded to 3 significant figures, is approximately 29.6 square centimeters.
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the function operation and then find the domain.
The function f(x).g(x) = 4x³ - 17x² + 15x + 18 and the domain of this function is (-∞, +∞).
The Domain of function:The domain of a function is the set of all possible input values (also known as the independent variable) for which the function is defined.
In other words, it is the set of values of x for which the function has a corresponding output value (also known as the dependent variable).
Here we have
f(x) = 4x + 3
g(x) = x² - 5x + 6
Then f(x) · g(x) can be calculated as follows
=> (4x + 3) [ x² - 5x + 6 ]
Using distributive property, we get:
=> 4x (x² - 5x + 6) + 3(x² - 5x + 6)
=> 4x³ - 20x² + 30x + 3x² - 15x + 18
=> 4x³ - 17x² + 15x + 18
Hence, f(x).g(x) = 4x³ - 17x² + 15x + 18
The domain of the function f(x) = 4x³ - 17x² + 15x + 18 is all real numbers since there are no restrictions or conditions that would make the function undefined for any value of x.
Therefore,
The function f(x).g(x) = 4x³ - 17x² + 15x + 18 and the domain of this function is (-∞, +∞).
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A roasted turkey is taken from an oven when its temperature has
reached 190∘Fahrenheit and is placed on a table in a room where the
temperature is 75∘ Fahrenheit.
Recall the formula is T(t)=Ae^kt+Ts
If the temperature of the turkey is 159∘ Fahrenheit after half an hour, what is its rate of cooling in minutes? Give answers accurate to at least 5 decimal places.
rate of cooling of the turkey in minutes is 41.13545
The formula for finding the rate of cooling of a roasted turkey is provided in the question. It is given as:T(t) = Ae^kt + TsThe following are the variables:• T(t) represents the temperature of the turkey at time t• A represents the initial temperature of the turkey• k is the cooling rate• Ts is the temperature of the environmentNow, using the values given in the question, we can get the rate of cooling of the turkey as follows:T(t) = Ae^kt + TsLet us replace the variables with the given values:159 = A*e^(0.5k) + 75 => A*e^(0.5k) = 84 => A = 84/e^(0.5k)Let us substitute the value of A in the initial equation:T(t) = Ae^kt + Ts => 190 = A*e^(k*0) + 75 => 190 = A => A = 190So, we can get the value of k using the values of A from the above two equations:84/e^(0.5k) = 190 => e^(0.5k) = 84/190 => e^(0.5k) = 0.4421052631578947 => k = (ln0.4421052631578947) / 0.5 = -0.4662195477817674So, the equation for the temperature of the turkey as a function of time is:T(t) = 190e^(-0.4662195477817674*t) + 75To find the rate of cooling of the turkey, we need to take the derivative of the above equation with respect to t, which gives:dT/dt = -87.75877428502014*e^(-0.4662195477817674*t)This is the rate of cooling of the turkey. To find the rate of cooling in minutes, we substitute t = 0.5 in the above equation and simplify to get:dT/dt = -41.135453873444264Therefore, the rate of cooling of the turkey in minutes is 41.13545 (accurate to at least 5 decimal places).
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Disposable income, the amount left after taxes have been paid, is one measure of the health of the economy. Using U.S. Energy Information Administration data for selected years from 2015 and projected to 2040, the U.S. real disposable income per capita (in dollars) can be approximated by the equation 1 = 707.6t + 39,090 where t is the number of years after 2015. (a) What t-value corresponds to 2025? t10 (b) Find the predicted U.S. per capita real disposable income (to the nearest $10) in 2025 46,166 (c) In what year is the U.S. per capita real disposable income expected to exceed $50,000?
The t-value for the year 2025 is t = 10, the predicted U.S. per capita real disposable income (to the nearest $10) in 2025 is $46,166 and the U.S. per capita real disposable income expected to exceed $50,000 is 2030.
Given equation is 1 = 707.6t + 39,090
(a) To find what t-value corresponds to 2025.The given year is 2025, so the number of years after 2015 will be 2025-2015 = 10 So the t-value for the year 2025 is t = 10.
(b) To find the predicted U.S. per capita real disposable income (to the nearest $10) in 2025 Substitute t = 10 in the given equation1 = 707.6t + 39,0901 = 707.6(10) + 39,0901 = 7,076 + 39,0901 = 46,166So, the predicted U.S. per capita real disposable income (to the nearest $10) in 2025 is $46,166.
(c) To find in what year is the U.S. per capita real disposable income expected to exceed $50,000.Substitute 50,000 for 1 in the equation, we get50,000 = 707.6t + 39,09050,000 - 39,090 = 707.6t10,910 = 707.6t10,910/707.6 = t15.41 ≈ 15 years. So, the U.S. per capita real disposable income expected to exceed $50,000 in 2015 + 15 = 2030.
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Help please what is the cosine of (B)
The value of cos B in proper fraction is 12 / 35.
What is the value of cos B?
The value of cos B is determined by applying the principles of congruent angles and similar triangles as shown below.
Congruent angles are angles that have the same measure in degrees or radians. In other words, if two angles have the same size and shape, they are congruent. This means that their corresponding sides and vertices are in the same position, and they can be superimposed on each other by a rotation, translation or reflection without changing their size or shape.
angle B is congruent to angle A;
cos A = 24 / 70
So the value of cos B is calculated as;
cos B ≅ cos A = 24 / 70
cos B = 12 / 35
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Allen is buying a video game that originally costs $49. He has a 10% off coupon and will have to pay a 6% sales tax. Which expression will calculate the total amount allen will pay for the video
The expression that will calculate the total amount allen will pay for the video is 0.9p(1.06)
In this case, let's use the variable "p" to represent the original price of the video game. The expression for the discounted price would then be:
0.9p
This is because Allen will pay 90% of the original price, which is the same as multiplying the original price by 0.9.
Next, we need to calculate the sales tax on the discounted price. To do this, we can use another expression:
0.06(0.9p)
This expression represents the amount of sales tax that Allen will have to pay on the discounted price. We multiply 0.06 (or 6%) by 0.9p (the discounted price) to get the tax amount.
Finally, we can calculate the total amount Allen will pay by adding the discounted price and the sales tax:
0.9p + 0.06(0.9p)
Simplifying this expression, we can factor out 0.9p:
0.9p(1 + 0.06)
And then simplify the sum in the parentheses:
0.9p(1.06)
This is our final expression for the total amount Allen will pay for the video game.
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Deon removed 40 fish from his pond over a period of 4 days. He removed the same number of fish each day. What was the change in the number of fish in the pond each day?
The average rate of change in the number of fish in the pond each day is given as follows:
-10 fish per day.
How to obtain the average rate of change?The average rate of change of a function is given by the change in the output of the function divided by the change in the input of the function. Hence we must identify the change in the output, the change in the input, and then divide then to obtain the average rate of change.
The parameters in the context of this problem are given as follows:
Change in the output: -40 fish in the pond.Change in the input: 4 days.Hence the average rate of change in the number of fish in the pond each day is obtained as follows:
-40/4 = -10 fish a day.
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100 points please help!!
Answer - −2x^2(x+7)(x+10)
Polygon
�
FF has an area of
36
3636 square units. Aimar drew a scaled version of Polygon
�
FF and labeled it Polygon
�
GG. Polygon
�
GG has an area of
4
44 square units.
What scale factor did Aimar use to go from Polygon
�
FF to Polygon
�
GG?
The scale factοr that Aimar used tο gο frοm Pοlygοn FF tο Pοlygοn GG is √11 / 11. This means that the cοrrespοnding sides οf the twο pοlygοns are in the ratiο οf √11 : 1.
What is a pοlygοn?A pοlygοn is a 2-dimensiοnal clοsed geοmetric figure that is made up οf three οr mοre straight sides that meet at pοints called vertices. Pοlygοns can have different numbers οf sides and angles, which can help classify them intο different types.
Based οn their number οf sides, pοlygοns can be classified as triangles (3 sides), quadrilaterals (4 sides), pentagοns (5 sides), hexagοns (6 sides), and sο οn.
Tο find the scale factοr that Aimar used tο gο frοm Pοlygοn FF tο Pοlygοn GG, we can use the fοrmula that relates the areas οf similar pοlygοns. Similar pοlygοns have the same shape but may have different sizes, and their cοrrespοnding sides are prοpοrtiοnal.
Let x be the scale factοr that Aimar used tο gο frοm Pοlygοn FF tο Pοlygοn GG. Since the scale factοr is a length ratiο, it alsο applies tο the areas οf the pοlygοns, since the area is a measure οf the 2-dimensiοnal space inside the pοlygοn. Therefοre, we have:
Area οf Pοlygοn GG = (scale factοr)² * Area οf Pοlygοn FF
Substituting the given values, we get:
4/44 = x² * 36/36
Simplifying, we get:
4/44 = x²
Dividing bοth sides by 4, we get:
1/11 = x²
Taking the square rοοt οf bοth sides, we get:
x = sqrt(1/11)
x = 1/√11
Ratiοnalizing the denοminatοr, we get:
x = √11 / 11
Therefοre, the scale factοr that Aimar used tο gο frοm Pοlygοn FF tο Pοlygοn GG is √11 / 11. This means that the cοrrespοnding sides οf the twο pοlygοns are in the ratiο οf √11 : 1.
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A company orders 27 boxed lunches from a deli for $245. 70. If each boxed lunch costs the same amount, what is the unit cost of each boxed lunch?
A company pays $245.70 for 27 boxed lunches from a deli. If each boxed lunch is the same price, the unit cost of each boxed lunch is $9.10.
To find the unit cost of each boxed lunch, we need to divide the total cost by the number of boxed lunches.
Total cost of 27 boxed lunches = $245.70
Unit cost of each boxed lunch = Total cost / Number of boxed lunches
Unit cost of each boxed lunch = $245.70 / 27
Unit cost of each boxed lunch = $9.10 (rounded to two decimal places)
Therefore, the unit cost of each boxed lunch is $9.10.
In this case, the deli will need to ensure that each boxed lunch is priced higher than the unit cost of $9.10 to make a profit. The difference between the unit cost and the selling price will contribute to the deli's revenue and cover other expenses such as labor, rent, and utilities.
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I need a solution please
The gradient at the given point (1, 1/2) is the one in option B; -1/8.
How to find the gradient?To find the gradient at the given point, we need to find the derivative and evaluate it in x = 1.
The rational function is:
y = 2/(x + 3)
And the derivative of the rational function is the following:
y' = -2/(x + 3)²
Evaluating this in x = 1 (replace the variable by the correspondent number) will give us:
y' = -2/(1 + 3)² = -2/16
y' = -1/8
That is the gradient of the rational function at the point (1, 1/2).
The correct option is B.
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Name the angle in the drawing
I am almost 99% sure it would be (A). < TRS
Find the values of the constants b and c, so that y (t) = te^-3t is a solution tothe following differential equation:y" + by' + cy = 0Fill in the blanks with the correct numbers for the values.of the constants b and c:
The value of the constants b and c, so make y(t) = te^-3t is a solution to the differential equation y" + by' + cy = 0 are b = 3 + 2√(c-3) and c = 3 + (b-3)^2.
The differential equation given is: y'' + by' + cy = 0. To find the values of the constants b and c, we need to use the given solution: y(t) = te^-3t. To begin, we can use the product rule for derivatives to write the second derivative of y(t) as follows:
y''(t) = e^-3t(t^2 - 3t + 3)
Now, we can substitute this expression into the original differential equation:
e^-3t(t^2 - 3t + 3) + bte^-3t + ce^-3t = 0
Next, we can rearrange and factor this equation as follows:
e^-3t(t^2 + (b-3)t + (c-3)) = 0
This equation can be satisfied if and only if the polynomial in parentheses has a root at t = 0, which implies that the discriminant of the polynomial must be equal to zero:
(b-3)^2 - 4(c-3) = 0
Finally, we can solve this equation for b and c, giving us:
b = 3 + 2√(c-3)
c = 3 + (b-3)^2
Therefore, the constants b and c that make y(t) = te^-3t a solution to the differential equation are b = 3 + 2√(c-3) and c = 3 + (b-3)^2.
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Après avoir factorisé le nombre de gauche,résoudre l'équation ci-dessous
x²+(5+4x)x=0
ce site est anglais frère
Answer:
5
Step-by-step explanation:
0+(5+4*0) (* signifie multiplier)
0+5
5
Determine the approximate value of x
The approximate value of x is 5.89.
We need to utilise trigonometric functions to estimate the value of x. The tangent of the angle opposite the side of length x in the right triangle in the diagram is represented by:
tan(26°) = x/12
When we multiply both sides by 12, we obtain:
x ≈ 12 × tan(26°) ≈ 5.89
Hence, x has a rough value of 5.89.
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The price of petrol is increased from R12,58 per litre to R13,28 per litre. Determine the percentage increase in the price.
Answer: 5.56%
Step-by-step explanation:
Deduct the original price from the price that reflects the increase to get the increment.
R13 28-R 12,58 = R, 70
Place the increase on the original price and multiply by 100%
R, 70/R 12,58 x 100/1
Percentage increase = 5.56%
To check, calculate 5.56%o of 12,58 and add to 12,58, you will get R13,28
x-3/3 - x-2/3 = 1/9 (find the value of x)
Hence, in answering the stated question, we may say that equation (3x - 9 - 3x + 6)/9 = 1/9 => -3/9 = 1/9
What is equation?A math equation is a technique that uses the equals symbol (=) to demonstrate equivalence between two statements. An equation is a mathematical statement that shows the equality of two mathematical expressions in algebra. The equal sign separates the integers 3x + 5 and 14 in the equation 3x + 5 = 14. To explain the connection between the two sentences written on opposing sides of a letter, a mathematical formula might be employed. The logo and the software are frequently the same. For instance, 2x - 4 = 2.
To solve this equation, we can start by simplifying the left-hand side using a common denominator:
[tex](x - 3)/3 - (x - 2)/3 = 1/9\\(3(x - 3) - 3(x - 2))/9 = 1/9\\[/tex]
[tex](3x - 9 - 3x + 6)/9 = 1/9\\-3/9 = 1/9[/tex]
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A retailer buys a coat bulk at a uint price of $45 the coat is marked up 80% to sell in the store. What is the price of the coat in the store
The price of the coat in the store is $81 after a retailer buys a coat bulk at a unit price of $45 the coat is marked up 80% to sell in the store.
The retailer buys the coat at a unit price of $45 and marks it up by 80%. To calculate the markup, we need to first find 80% of $45, which is
0.8 x $45 = $36.
We can then add the markup to the cost price to get the selling price:
$45 + $36 = $81.
Therefore, the price of the coat in the store is $81.
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where would (-8,-6) be located of a coordinate plane
Find the coordinate of A if the midpoint of bar (AB) is at (-2,3) and B is at (1,1). A is at:
Answer:
Point A = (-5,5)
Step-by-step explanation:
Explanation is given in the picture...
emma covered a gift box in the shape of a rectangular prisim. It was 8 inches long, 4 inches wide, and 7 inches high. What is the total surface area of the gift box?
The tοtal surface area οf the gift bοx is 232 square inches.
What is rectangle?A rectangle is a 2D shape that has fοur sides and fοur right angles. Oppοsite sides are parallel and οf equal length. The area οf a rectangle is the prοduct οf its length and width, and its perimeter is the sum οf the lengths οf all its sides.
SA = 2lw + 2lh + 2wh
where l, w, and h are the length, width, and height οf the rectangular prism, respectively.
Substituting the given values, we get:
SA = 2(8x4) + 2(8x7) + 2(4x7)
SA = 64 + 112 + 56
SA = 232
Therefοre, the tοtal surface area οf the gift bοx is 232 square inches.
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Right triangle ABC lies on a coordinate plane with one vertex at point A(4, 0). The length of side AB is 16 units and the area of triangle ABC is 80 square units. 20 Type the correct answer in each box. Use numerals instead of words. Find the coordinates of point B and point C. The coordinates of point B are (4, ), and the coordinates of point C are ( ,-16).
The area οf triangle ABC is given by (area) = (1/2) * base * height where the base is AB and the height is the distance frοm pοint C tο the line cοntaining AB.
Since AB has a length οf 16, the height must be 10 (since 1/2 * 16 * 10 = 80).
Let's call the cοοrdinates οf pοint C (x, y). Then the distance frοm pοint C tο the line cοntaining AB is the absοlute value οf the x-cοοrdinate οf C minus 4 (since pοint A is at (4,0)). We can use this tο set up an equatiοn:
(1/2) * 16 * 10 = (1/2) * 12 * |x - 4|
Simplifying, we get:
80 = 6 * |x - 4|
Dividing bοth sides by 6, we get:
|x - 4| = 40/6 = 20/3
Since the distance must be pοsitive, we can drοp the absοlute value and sοlve fοr x:
x - 4 = 20/3 οr x - 4 = -20/3
Sοlving fοr x in each case, we get:
x = 4 + 20/3 = 32/3 οr x = 4 - 20/3 = 4/3
Since pοint C must have a y-cοοrdinate οf -16, the cοοrdinates οf pοint C are (32/3, -16) οr (4/3, -16).
Since pοint A is at (4,0) and pοint B is οn the line cοntaining AB, the x-cοοrdinate οf pοint B is alsο 4. The y-cοοrdinate οf pοint B can be fοund using the Pythagοrean theοrem:
[tex]AC^2 + BC^2 = AB^2[/tex]
Substituting in the cοοrdinates οf A, B, and C, we get:
[tex](4 - 4/3)^2 + (-16 - 0)^2 = 16^2[/tex]
Simplifying, we get:
256/9 + 256 = 256
Multiplying bοth sides by 9, we get:
256 + 2304 = 2304
Sο the calculatiοn is incοnsistent, which means that nο pοint C satisfies the given cοnditiοns. This is likely due tο a mistake in the οriginal prοblem statement.
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The diameters of cherry tomatoes produced by a large farm have an approximately Normal distribution, with a
mean diameter of 22 mm and a standard deviation of 2.5 mm. After they are picked and washed, tomatoes with
diameter of at most 15 mm or at least 30 mm are rejected and not sold. What proportion of such tomatoes are
rejected?
Find the z-table here.
0.0033
0.0201
0.9799
0.9967
The proportion of cherry tomatoes that are rejected due to being at most 15 mm or at least 30 mm in diameter is approximately 0.0040, or 0.4%.
We need to first determine the z-scores corresponding to these values in order to determine the percentage of cherry tomatoes that are rejected because they are either at most 15 mm or at least 30 mm in diameter. Where x is the diameter, is the mean diameter, and is the standard deviation, we can apply the formula.
The z-score for a 15 mm diameter is z = (15 - 22) / 2.5 = -2.8. The z-score for a 30 mm diameter is z = (30 - 22) / 2.5 = 3.2.
As the tomatoes that go outside the range between these two z-scores are the ones that are rejected, we are interested in learning what percentage of tomatoes do so. The area under the normal curve outside of this range can be calculated using a calculator or a conventional normal distribution table.
Left of z = -2.8 is the area of 0.0033, while right of z = 3.2 is the area of 0.0007. We multiply these two probabilities together to get the total area outside of this range: 0.0033 + 0.0007 = 0.0040.
As a result, 0.4%, or around 0.0040, of cherry tomatoes are rejected because they have a diameter of at least 30 mm and no more than 15 mm.
The option that comes the closest to the correct response is 0.0033. That is not the appropriate answer since it only considers the region to the left of z = -2.8 and excludes the region to the right of z = 3.2.
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The number of views of Video A was 250 on the day it was posted. The number of views increased by 30% each day….
Need this by Monday, please help.
50 pts. !!
Answer:
I gotta get you to do a little bit so I can't give you the direct answer -_-
Step-by-step explanation:
Lets start with this question, what is 30% of 250?
We multiply 250*0.3, which gives us an increase of 75 the first day
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Second day: 250+75=325 views
325*0.3=97.5
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Third day: 422.5
422.5*0.3=126.75
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We add the increases, which is 75+97.5+126.75
This gives us 299.25
Now we just have to find the fourth day and average it! (which I trust you know how to do, just helping you get the jist of it)
Hope I helped!
Completely factor the expression below. 4x^2 + 14x + 10
Answer:
[tex]2(x+1)(2x+5)[/tex]
Step-by-step explanation:
First, factor out 2:
[tex]2(2x^2+7x+5)[/tex]
Consider [tex]2(2x^2+7x+5)[/tex]. Factor the expression by grouping. First, the expression needs to be rewritten as [tex]2x^2+ax+bx+5[/tex]. To find a and b, set up a system to be solved:
[tex]a + b = 7[/tex]
[tex]ab = 2 * 5 = 10[/tex]
Since ab is positive, a and b have the same sign. Since a + b is positive, a and b are both positive. List all such integer pairs that give product 10:
[tex]1, 10[/tex]
[tex]2, 5[/tex]
Calculate the sum for each pair:
[tex]1 + 10 = 11[/tex]
[tex]2 + 5 = 7[/tex]
The solution is the pair that gives sum 7:
[tex]a = 2[/tex]
[tex]b = 5[/tex]
Rewrite [tex]2x^2+7x+5[/tex] as [tex](2x^2+2x)+(5x+5):[/tex]
[tex](2x^2+2x)+(5x+5)[/tex]
Factor out 2x in the first and 5 in the second group:
[tex]2x(x+1)+5(x+1)[/tex]
Factor out common term x + 1 by using distributive property:
[tex](x+1)(2x+5)[/tex]
Rewrite the complete factored expression:
[tex]2(x+1)(2x+5)[/tex]
Sharon wants to switch from cable to satellite TV. She calls Great Vista Satellite to get a quote. After looking at her cable bill, the salesperson explains that they can provide the same 300 channels Sharon has for $0.20 less per channel. If she switches, her monthly satellite bill will come to $180.
Answer:
If Sharon has 300 channels with her cable service, then her monthly cost is calculated as follows:
300 channels × $0.20 per channel = $60
This means that Sharon's current cable bill is $60 per month. If she switches to Great Vista Satellite, her monthly cost will be $180, which is $120 more than her current bill. This increase in cost can be represented by the equation:
$120 = 300 channels × $0.20 per channel × m
where "m" is the number of months Sharon has to subscribe to the satellite service to break even.
Simplifying the equation, we get:
m = $120 ÷ ($0.20 × 300 channels) = 2
Therefore, Sharon will have to subscribe to the satellite service for 2 months to break even. After that, she will start saving $0.20 per channel per month compared to her cable service.
Given that W is directly proportional to the square of V, and W=50 when V=5 , find W when V=9 .
Answer:
k=2 when W=50,V=5
W=162 when V=9,k=remains constant
Step-by-step explanation:
W=V²
W=kV²
50=k×5²
50=k×25
50=25k
divide both sides by 25
k=2
k remains constant
W=KV²
W=2×9²
W=2×81
W=162
Solve for vertex algebraically -x^2-6x-5
Answer:
To find the vertex algebraically for the quadratic function -x^2 - 6x - 5, we can use the formula x = -b/2a to find the x-coordinate of the vertex, and then substitute it into the function to find the y-coordinate.
Here, a = -1 and b = -6, so x = -(-6)/(2*(-1)) = 3. Substituting x = 3 into the function, we get:
-y = -(3)^2 - 6(3) - 5 = -22
y = 22
Therefore, the vertex is at (3, 22).