The coordinates of the vertex for f(x) = 3x²+6x+3 are (-1, 0).
For any equation of the type y=ax²+bx+c, the vertex is given by (h, k)
Where, h = -b/2a and k = (4ac - b²)/4a.
Given, the equation is y = 3x²+ 6x+3
Here, a = 3, b = 6 and c = 3
So, h = -6/2(3)
h = -6/6
h = -1
4ac = 4(3)(3)
4ac = 36
So, k = [36 - (6)2]/4(3)
k = (36 - 36)/12
k = 0
Therefore, the coordinates of the vertex are (h,k) = (-1, 0).
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How do you find the median of grouped data class 10 example?.
To find the median of grouped data if (n) is odd the median is (n+1)/2 and if (n) is even the median will be average of n/2th and the (n/2)+1.
Data that is written continuously and is sorted in ascending order is the median of a bunch of data. The information is presented as a frequency distribution table that separates higher level information from lower level information. Using the formula is one of the simplest ways to determine the median of grouped data.
We can employ the procedures and formula below to determine the median for grouped data:
The complete number of observations can be found.Set the class size and create various classes for the data.Determine the overall frequency for each class.Determine the category that the median belongs to. (N/2 lies in the class defined by the median.)Find the cumulative frequency of the median class as well as the lower limit of the median class (l) (c).Learn more about Grouped Data:
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What is a term formula?.
The formula is a fact or a rule written with mathematical symbols. It usually connects two or more quantities with an equal sign.
In arithmetic, a chain is an enumerated series of gadgets in which repetitions are allowed and order matters. Like a hard and fast, it consists of contributors (additionally referred to as elements, or terms). The quantity of elements (possibly endless) is called the duration of the series.
Unlike a fixed, the same factors can appear a couple of times at distinct positions in a chain, and unlike a fixed, the order does be counted. formally, a chain can be described as a function from herbal numbers (the positions of elements inside the collection) to the factors at each position. The belief of a sequence may be generalized to a listed family, defined as a character from an arbitrary index set.
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What is inverse graph in graph theory?.
If the graphs of both functions are symmetric with respect to the line y = x, then we say that the two functions are inverses of each other.
The inverse function is represented by f⁻¹ with regards to the original function f and the domain of the original function becomes the range of the inverse function.
The range of the given function becomes the domain of the inverse function the injective function is the reflection of the original function with reference to the line y = x and is obtained by swapping (x, y) with the (y, x).
There are a few steps to find an Inverse Function:
For the given function f(x) = ax+b, replace f(x)=y, to obtain y = ax + b.Interchange the x with y and the y with x in the function y=ax+b to obtain x=ay+b.Here solve the expression x = ay + b for y. And we obtain y = (x - b/aFinlly replace y = f-1(x), and we have f-1(x) = (x - b)/a.To know more about inverse graph function:
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Which ordered pair i a olution to the ytem of inequalitie?
x y > −2
−x y < 1
The solution to the system of inequalities given by x + y > −2 and −x + y < 1 is the ordered pair that lies under the conditions x > -3/2 and y > -1/2. One of which is (1, 0).
An inequality is a statement about the relative size or order of two values. It's a statement that one value is greater than, less than, greater than or equal to, less than or equal to, or not equal to another value. It's represented using mathematical symbols such as <, >, ≤, ≥, ≠.
To find the solution to a system of inequalities, rewrite any of the two inequalities as y in terms of x.
x + y > -2 can be rewritten as y > -x - 2
Substitute the value of y into the other inequality.
-x + y < 1 will become -x + (-x - 2) < 1
Solve for the value of x.
-x + (-x - 2) < 1
-2x -2 < 1
-2x < 1 + 2
-2x < 3
2x > -3
x > -3/2
Plug in the value of x into the equation of y and solve for y.
y > -x - 2
y > -(-3/2) - 2
y > -1/2
So, the solution of the system of inequalities will be the region where both of these conditions, x > -3/2 and y > -1/2, are true, which is the area where the graph of y > -x - 2 and y < x + 1 overlap.
One ordered pair (1 , 0) is a solution to the system of inequalities x + y > -2 and -x + y < 1.
We can check it by plugging in the ordered pair into the inequalities.
x + y > -2 becomes 1 + 0 > -2 which is true
-x + y < 1 becomes -1 + 0 < 1 which is true
So (1,0) satisfies both inequalities.
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94% of ticket old at an amuement park were dicount ticket. If the park old 9,000 ticket in all, how many dicount ticket did it ell
The total discount tickets sold are 8460 tickets.
A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.
The discount equals the difference between the price paid for and it's par value. Discount is a kind of reduction or deduction in the cost price of a product. It is mostly used in consumer transactions, where people are provided with discounts on various products. The discount rate is given in percentage.
We have that,
94% = 94/100
= 0.94
By multiplying the tickets sold by the factor of 0.94, we get
0.94 × 9000
8460 tickets
The total discount tickets sold are 8460 tickets.
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Which expression is equivalent to
(-7 + 2i) + 6i- (11-15) ?
Select one:
O a.-18 + 23i
O b.-18-7i
O c. -4 +23i
O d. -4-7i
The equivalent expression of (-7 + 2i) + 6i- (11-15) is -3 + 8i.
How to find equivalent expression?Two algebraic expressions are said to be equivalent if their values obtained by substituting the values of the variables are same. In other words, equivalent expression are usually the same.
We can know the equivalent expression of (-7 + 2i) + 6i- (11-15) by simplifying it.
Hence, let's simplify (-7 + 2i) + 6i - (11 - 15) as follows:
Therefore,
(-7 + 2i) + 6i - (11 - 15) = -7 + 2i + 6i - 11 + 15
combine like terms
-7 + 2i + 6i - 11 + 15 = -7 + 15 - 11 + 2i + 6i
Hence,
-7 + 15 - 11 + 2i + 6i = -3 + 8i
Therefore,
(-7 + 2i) + 6i - (11 - 15) = -3 + 8i
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I need help on this question please And thank you7-5x+1
Answer: -5x+8
Step-by-step explanation:
add the like terms
Answer:
-5x+8
Step-by-step explanation:
You just have to isolate the 5x, and it has a "-" before it, so the 5x is a negative expression (-5x), then you do 7+1 that is 8.
A parabolic tunnel has a width of 20 feet and a height of 12 feet at the center. Find the cross-sectional area of the parabolic tunnel.
A. Illustrate the parabolic tunnel. Show the measurements.
B. What function represents the parabolic tunnel? Show your solution.
C. Compute for the cross-sectional area of the parabolic tunnel. Use the Definite Integral. Show your solution.
No troll.
Answer:
A. See attached graph
B. [tex]\boxed{y = 0.12x^2+ 12}[/tex]
C. [tex]\boxed{160\;square\;feet}[/tex]
Step-by-step explanation:
Consider the figure which shows the cross-sectional view of the parabolic tunnel as viewed from the entrance
A. See attached graph
B. The parabola is drawn with the y-axis as the axis of symmetry.
The x-intercepts are at (-10, 0) and (10, 0)
The vertex is at (0, 12)
The vertex form equation for a parabola is
y = a(x - h)² + k
where (h, k) is the vertex - the (x, y) coordinates of the vertex
Here h = 0, k = 12
So the parabola equation is:
y = a(x - 0)² + 12
y = ax² + 12
To find a, plug in one of the x-intercepts, say (10, 0)
We get
0 = a(10)² + 12
0 = 100a + 12
100a = -12
a = -12/100 = -0.12
Therefore the equation of the parabola is
[tex]\boxed{y = 0.12x^2+ 12}[/tex]
C. The cross-sectional area is given by the formula:
D. To find the area of the parabola between x = -10 and x = 10 which forms the base of the parabola, you can use the formula
[tex]A = \dfrac{2}{3}bh\\\\[/tex]
where b is the base length and h is the height
Using the values for b = 20 and h = 12 we get
[tex]A = \dfrac{2}{3}\cdot 20 \cdot 12 = 160 \textrm { square feet}[/tex]
Using the definite integral, let's find the area between x = -10 and x = 10
I am not going through every step but leading you to the final answer
[tex]\int _{-10}^{10}\left(\:-0.12x^{2\:}+\:12\right)dx\\\\\\\textrm{We have}\\\\0.12\cdot \int _{-10}^{10}x^2dx=0.12\left[\frac{x^3}{3}\right]_{-10}^{10}\\\\[/tex]
[tex]=0.04\left[x^3\right]_{-10}^{10}\\\\= 80\\\\[/tex]
Therefore
[tex]-0.12\cdot \int _{-10}^{10}x^2dx = -80[/tex]
[tex]\int _{-10}^{10}12dx = \left[12x\right]_{-10}^{10} = 240[/tex]
So total area = -80 + 240 = 160 square feet
Ans: Cross sectional area = [tex]\boxed{160\;square\;feet}[/tex]
A container of coffee i 1/6 full. The container contain 2/3 pound of coffee a. Write a diviion expreion that repreent the capacity of the container. B. Write a related multiplication expreion that repreent the capacity of
the container. C. Find the capacity of the container
The division expression that represents the capacity of the container is 2/3 ÷ 1/6, the multiplication expression that represents the capacity of the container is 2 x 6 / 3, the capacity of the container is 4 pound.
An expression in math is a sentence with a minimum of two numbers or variables and at least one math operation. This math operation can be addition, subtraction, multiplication, or division.
Let us assume total capacity of the container = x pounds.
Container contains 1/6th coffee of total x pounds, that is x/6 pounds.
We also given that 1/6th part is equal to 2/3 of a pound.
So, we could setup an equation now.
x/6 pounds equals 2/3 of a pound.
[tex]\frac{x}{6} = \frac{2}{3}[/tex]
Dividing both sides by 1/6, we get
[tex]\frac{x}{6}[/tex] ÷ [tex]\frac{1}{6}[/tex] = [tex]\frac{2}{3}[/tex] ÷ [tex]\frac{1}{6}[/tex]
Therefore, x = [tex]\frac{2}{3}[/tex] ÷ [tex]\frac{1}{6}[/tex]
for multiplication expression,
x = [tex]\frac{2}{3}[/tex] × 6 { ∵ a ÷ b = a × [tex]\frac{1}{b}[/tex] }
the capacity of the container is,
x = [tex]\frac{2}{3}[/tex] × 6
x = 4 pound.
Therefore, the division expression that represents the capacity of the container is 2/3 ÷ 1/6, the multiplication expression that represents the capacity of the container is 2 x 6 / 3, the capacity of the container is 4 pound.
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What is the difference between the smallest 6 digit number with five different digits?.
The difference between the smallest 6-digit number with five different digits and the greatest 5-digit number with four different digits is 358.
The aim is to find the largest 5-digit number with four different digits and the smallest 6-digit number with five different digits, and then we need to find how many numbers are between these two values.
Let's start by finding the smallest 6-digit number with five different digits.
We know that 0 is the smallest digit that can not be used in the highest place value, so we will consider the next smallest digit, 1; therefore, it can be used in the highest place value.
Thus, we can create the required 6-digit smallest number with 0 and 1.
Hence, the smallest 6-digit number is 100234.
Now, we will discover the largest 5-digit number.
As we know, the largest digit is 9, which can be used anywhere. So, we will form the largest 5-digit number with the digit 9.
Thus, the largest 5-digit number is 99876.
Now, calculate the difference between these two numbers.
So, we can determine the difference between the smallest 6-digit number and the largest 5-digit number by subtracting the smallest from the largest.
Thus, we get,
⇒100234 - 99876
⇒358
Hence, 358 is the difference between the smallest 6-digit with five different digits and the largest 5-digit number with four different digits.
--The given question is incomplete; the complete question is
"What is the difference between the smallest 6-digit number with five different digits and the greatest 5-digit number with four different digits?"--
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Which equation represents the horizontal line passing through (-3, 6)?
The equation of the line passing through (-3, 6) is y = 6.
What is an equation?An equation is a mathematical statement that shows that two mathematical expressions are equal.
Given that, a horizontal line is passing through (-3, 6), we are asked to find its equation,
Since, the equation that represents a horizontal line is of form y = a i.e, it must pass through point (0,a) on the y-axis.
We are given, the line passes through (-3, 6), it must pass through point (0,6) on the y-axis.
Hence, the equation of the line passing through (-3, 6) is y = 6.
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Good morning I need this done by an hour thanks!
how would you solve the following equation: x - 10 = 90
Answer:
x = 100
Step-by-step explanation:
Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. In this case, you will add 10 to both sides of the equation, to isolate x:
x - 10 = 90
x - 10 (+10) = 90 (+10)
x = 90 + 10
x = 100
x = 100 is your answer to the equation.
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urgant !!! help me plsease !!!
Answer: 370 cubic centimeters
Step-by-step explanation:
They're just boxes stack on top of another. Find volume of both box and add together.
Use the cuboid formula to find volume of each box:
[tex]Volume = length *width * height[/tex]
Bottom box volume: [tex]10cm*5cm*5cm=250^3cm[/tex]
Top box volume: [tex]8cm*5cm*3cm=120^3cm[/tex]
Total volume = [tex]250^3cm+120^3cm=370^3cm[/tex]
Why is triangle 180 degrees?.
A triangle is 180 degrees because the sum of the interior angles of any polygon is always equal to 180 degrees times the number of sides minus two.
A triangle is 180 degrees because the sum of the interior angles of any polygon (a closed shape with at least three straight sides) is always equal to 180 degrees times the number of sides minus two. This is known as the polygon angle sum theorem. In the case of a triangle, which has three sides, the sum of the interior angles is 180 degrees times one less than the number of sides, or 180 degrees times (3-2), which equals 180 degrees. Additionally, since a triangle is a three-sided polygon, and each angle in a triangle is connected by three straight lines, the sum of the three angles in a triangle must always add up to 180 degrees.
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Is F G x )) the inverse of g/f x ))?.
No, "FG)(x)" and "g/f)(x)" are not inverse of each other."FG)(x)" refers to the composition of two functions, "F" and "G", where the output of "G" is used as the input for "F".
The notation for this is represented as "F(G(x))", where "x" is the input for both functions.
The function "F" is applied to the output of the function "G", meaning that the output of the function "G" is used as the input for the function "F".
On the other hand, "g/f)(x)" is not a standard notation in mathematics. It could be interpreted as applying the function "g" to the input "x", and then applying the function "f" to the result of "g". But it could also be interpreted as applying f to the input x and then applying g to the result of f.
Inverse functions are related to the idea of reversing the effect of a function. For example, if "f" is a function that maps a set of inputs to a set of outputs, then the inverse function "f^-1" maps the set of outputs back to the original set of inputs. The notation for an inverse function is typically represented as "f^-1(x)" or "f^(-1)(x)". It's important to note that not all functions have inverses, and the inverse of a function is also a function.
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Katie rolled a fair number cube, with face numbered 1 to 6, 200 time. She rolled the number 5, five time. What i the experimental probability of landing on landing on 5? How doe it compare to the theoretical probability of landing on 5?
The theoretical likelihood of receiving a 2 is 1/6, but the actual likelihood is 1/3.
They are divided by one-half.
The potential likelihood
The theoretical likelihood of receiving a 2 is 1/6, which is the ratio of the number of fortunate events to all possible outcomes.
The experimental likelihood
The experimental probability of an event occurring is the proportion of total times the experiment was conducted that an event occurred during the experiment.
in this case, is 6/12 or 1/2.
1/2 - 1/6 = 3/6 - 1/6 = 2/6 = 1/3
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Why can't the denominator of a fraction be negative?
When you divide a fraction by a negative number, it doesn't matter what happens to the denominator because the entire result, or quotient, is the reverse of whatever the original fraction was.
What is the denominator?In mathematics, a denominator is the lowest number in a fraction that indicates how many equal parts are divided into a whole.
It is a fraction's divisor. In this case, the denominator is 4, thus there are four components overall.
The top number in a fraction is referred to as the numerator, while the bottom number is referred to as the denominator.
4/5, for instance, is a fraction.
What happens to the denominator of a fraction when you divide it by a negative number is unimportant because the entire result, or quotient, is the opposite of whatever the original fraction was.
Therefore, when you divide a fraction by a negative number, it doesn't matter what happens to the denominator because the entire result, or quotient, is the reverse of whatever the original fraction was.
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FIND THE THIRD,FIFTH AND TENTH TERM SEQUENCE DESCRIBED BY EACH RULE
A(n)=4+(N-1)(8)
A(3)=
In the given arithmetic series, the third, fifth, and tenth terms are, respectively, 20, 36, and 76.
What is an arithmetic sequence?The difference between every two successive terms in an arithmetic series is always the same. The number "a" is the first term, and "d" is the common difference between the. sequence. The nth term of an arithmetic sequence is given by. aₙ = a + (n – 1)d
Given, Term sequence described by each rule
A(n)=4+(N-1)(8)
thus,
Third term A(3) = 4 + (3 -1)8
A(3) = 20
Fifth term A(5) = 4 + (5 -1)8
A(5) = 36
tenth term A(10) = 4 + (10 - 1)8
A(10) = 76
Therefore, the third fifth and tenth terms of the given arithmetic series are 20, 36, and 76 respectively.
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Match a recursive formula for the graph and identify the sequence as arithmetic or geometric.
Graph of a sequence
Arithmetic, t0=15
and tn=2.8+tn−1
where n≥1
.
Geometric, t0=15
and tn=2.8⋅tn−1
where n≥1
.
Arithmetic, t0=15
and tn=2.8⋅tn−1
where n≥1
.
Geometric, t0=15
and tn=2.8+tn−1
where n≥1
.
The recursive formula in this case is;
Arithmetic, t0=15
and tn=2.8+tn−1
where n≥1
Option A
What is a recursive formula?A recursive formula is a formula that defines a sequence of values in terms of its previous values. It provides a way of defining a sequence in a compact form, with each term of the sequence depending on one or more preceding terms.
In mathematics, recursive formulas are often used to define sequences. We can see that in this sequence, the first term is 15 and the common difference is obtained by simple subtraction .
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Okay so say Jhonny had duties to complete for the company, each duty was 15 minutes each. If 4 duties equal an hour, and jhonny did 358 duties how many hours did he do in total?
Answer:
he did 89.5 hours in total
Step-by-step explanation:
Using umitary mwthod we get
4. 1
358. x
soling it
4x=358
or,x=358÷4
or,x=89.5 hours
©Thank You
Answered By Solution 49
The answer is 89.5 because it's 358 divided by 4.
What is the difference between exponential function and logarithmic function?.
Exponential functions and logarithmic functions are related but inverse of each other.
Exponential functions are used to model growth and decay, while logarithmic functions are used to model inverse growth and decay.
An exponential function is a function in the form of f(x) = b^x, where b is a positive constant called the base. The variable x is the exponent, and the value of the function is the result of raising the base to the power of x. For example, f(x) = 2^x is an exponential function. The graph of an exponential function is always increasing and never touches or crosses the x-axis, as long as b>1.
On the other hand, a logarithmic function is a function in the form of f(x) = log_b(x) or f(x) = ln(x), where b is a positive constant called the base (or e for natural logarithm) and x is the argument of the logarithm. The value of the function is the exponent to which the base must be raised to produce x. For example, f(x) = log2(x) is a logarithmic function. The graph of a logarithmic function is always decreasing and asymptotic to the x-axis.
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Is an acute angle 180?.
The angle 180 is not an acute angle.
We know that an angle is the figure, in which two rays meet at a common point (vertex).
The types of angles (based on their magnitude) are:
Acute Angle (angle measuring greater than 0° and less than 90°)
Obtuse Angle, (an angle which is less that 180° and greater than 90°)
and Right Angle (an angle that measures 90°)
There are some types of angle based on their rotation:
Straight Angle, (an angle measuring 180 degrees)
Reflex Angle, (an angle which is greater than 180 degrees but less than 360 degrees )
and Full Rotation (an angle measuring 360 degrees)
From the definition of acute angle,
acute angle measures greater than 0° and less than 90°
So, 180 degrees can not be an acute angle.
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At the beginning of the year, Salma had $35 in savings and saved an additional $8
each week thereafter. Henry started the year with $95 and saved $5 every week. Let
S represent the amount of money Salma has saved t weeks after the beginning of the
year and let H represent the amount of money Henry has saved t weeks after the
beginning of the year. Write an equation for each situation, in terms of t, and
determine the number of weeks after the beginning of the year until Salma and Henry
have the same amount of money saved. PLEASE ANSWERR
Answer:
20 weeks
Step-by-step explanation:
Salma's savings can be represented by the equation S = 35 + 8t, where t is the number of weeks after the beginning of the year.
Henry's savings can be represented by the equation H = 95 + 5t
To determine the number of weeks after the beginning of the year until Salma and Henry have the same amount of money saved, we need to set the two equations equal to each other and solve for t:
35 + 8t = 95 + 5t
3t = 60
t = 20
So it would take Salma and Henry 20 weeks after the beginning of the year to have the same amount of money saved.
HELP ME!!!
The photo below is the reduced form of a 16 by 20-inch print. What is the scale factor used to reduce it? Show Work.
The scale factor which was used to reduce the picture include the following: B. 1/4.
What is scale factor?In Geometry, a scale factor can be defined as the ratio of two corresponding side lengths or diameter in two similar geometric figures such as airplanes, which can be used to either horizontally or vertically enlarge (increase) or reduce (decrease or compress) a function that represents their size.
Mathematically, the scale factor of a geometric figure can be calculated by dividing the dimension of the image by the dimension of the pre-image (original figure):
Scale factor = Dimension of image/Dimension of pre-image
Scale factor = 4/16 = 5/20
Scale factor = 1/4.
Therefore, this photo was reduced by a scale factor of 1/4.
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Answer choices are:
4
1/4
12
1/12
Challenge The price of Stock A at 9 A.M. was $13.45. Since then, the price has been increasing at the rate of $0.08 each hour. At noon the price of Stock B was $14.20. It begins to decrease at the rate
of $0.14 each hour. If the two rates continue, in how many hours will the prices of the two stocks be the same?
In about hours, the prices of the two stocks will be the same.
(Round to the nearest tenth as needed.)
Answer:
Approximately 2.3 hours after 12 noon
Step-by-step explanation:
We are given
Stock A price at 9AM = $13.45
It increases by $0.08 per hour
Stock B at noon = $14.20
It decreases by $0.14 each hour
We have to find when the two prices will be the same.
Since the base time references are different - 9am for A and noon for B, we have to establish the same time reference
Let's do this by finding the price of stock A at noon, then we have the same time base reference of noon
9am - noon = 3 hours
In 3 hours stock A would have risen by 0.08 x 3 = $0.24
Price of stock A at noon is 13.45 + 0.24 = $13.69
Thereafter stock A continues to increase by the same rate of $0.08 per hour
The equation to model stock A price starting at noon is
13.69 + 0.08x where x is the number of hours elapsed after 12 noon
Stock B price is $14.20 at 12 noon and decreases by $0.14 per hour
Equation to model stock B price after 12 noon is
14.20 - 0.14x where x is the elapsed time after 12 noon
For both prices to be the same after the same x hours, we have
13.69 + 0.08x = 14.20 - 0.14x
Solving for x will tell us how many hours it will take for both stock prices to be the same
Solving
13.69 + 0.08x = 14.20 - 0.14x
So the two stock prices will be the same 2.3 hours after 12 noon or about 2 hours and 18 minutes after 12 noon or roughly around 2:18pm
I got 18 minutes by multiplying 0.3 x 60 = 18
A frozen yogurt company charges customers a flat fee of $1.00 plus $0.30 per ounce.
What are the 5 examples of solution?.
The 5 examples of solution are:
1. When we combine salt with water (often table salt), we create salt water. In this case, the solvent is water, and the solute is salt.
2. When sugar and water are combined, sugar water is created.
3. Several substances are dissolved in water to create mouthwash.
4. You may make iodine tincture by combining iodine crystals with alcohol.
5. In water, soda comprises sugar, carbon dioxide, color, and other ingredients.
Solvent and solute are combined to generate solutions. A solution is therefore a homogeneous mixture. Numerous solutions are available in our homes. Here is a short list:
1. Sugar and color are present in Kool Aid's water.
2. We make vinegar by combining acetic acid and water.
3. A solution of hydrogen peroxide between 3 and 6 percent is known as hydrogen peroxide solution.
4. This is created by combining water and hydrogen peroxide.
5. You can make detergent solution by combining detergent and water.
6. Window cleaner is made up of many chemicals in water with aroma.
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For what value of k will the following equations have infinite solutions 2x 3y 7?.
The value of k is 7
Consider the given equations are:
2x+3y=7
(k−1)x+(k+2)y=3k
The general equations
[tex]a_{1} x+b_{1} y =c_{1}[/tex]
[tex]a_{2} x+b_{2} y =c_{2}[/tex]
substituting the values,
[tex]a_{1} = 2, b_{1} y=3,c_{1}=7[/tex]
[tex]a_{2}=(k-1) ,b_{2} =( k+2), c_{2}= 3k[/tex]
An equation can have infinitely many solutions when it should assure some conditions. The system of an equation has infinitely numerous solutions when the lines are coincident, and they have the same y-intercept.
We know, the condition of infinite solutions
[tex]\frac{a_{1} }{a_{2} } = \frac{b_{1} }{b_{2} } = \frac{c_{1} }{c_{2} }[/tex]
Therefore,
[tex]\frac{2}{k-1} = \frac{3}{k+2} =\frac{7}{3k}[/tex]
⇒[tex]\frac{2}{k-1} =\frac{3}{k-2}[/tex]
⇒ 2k -4 = 3k-3
⇒ k = 7
Hence, the value of k is 7.
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The complete question is:
Find the value of k, infinitely many solutions
2x+3y=7,(k−1)x+(k+2)y=3k
A scientist mixes water (containing no salt) with a solution that contains 40% salt. she wants to obtain 120 ounces of a mixture that is 35% salt. how many ounces of water and how many ounces of the 40% salt solution should she use?
She must use 187.5 ounces of water and 62.5 ounces of salt solution for her experiment.
What is an equation?
The definition of an equation in algebra is a mathematical statement that proves two mathematical expressions are equal. Consider the equation 3x + 5 = 14, where 3x + 5 and 14 are two expressions that are separated by the symbol "equal."
The smallest equivalent fraction of the number is the form that is the simplest. How to find the most basic form: In the numerator and denominator, look for shared factors. Examine the fraction to see if one of the numbers is a prime number.
Say w= the amount of water in oz, then the 60% of the solution is 250 - w
Set up the equation:
Use distributive property first
0.60(250-w) = 0.15 x 250
Isolate to solve for w
150- 0.60w= 37.5
-0.60w= 37.5-150
0.60w= -112.5
w= 187.5 ounces
250-187.5= 62.5 ounces salt solution
Therefore, she must use 187.5 ounces of water and 62.5 ounces of salt solution for her experiment.
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