The length of unknown side x is 58.
The correct answer is option D.
To find the unknown side length, x, in a right triangle with the base measuring 42 and the perpendicular measuring 40, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let x be the hypotenuse. Applying the Pythagorean theorem, we have:
[tex]x^2 = 42^2 + 40^2[/tex]
Simplifying:
[tex]x^2[/tex] = 1764 + 1600
[tex]x^2[/tex]= 3364
Taking the square root of both sides to solve for x:
x = [tex]\sqrt{3364}[/tex]
Simplifying the square root:
x = ([tex]\sqrt{4 * 841)}[/tex]
Since 841 is a perfect square ([tex]29^2[/tex]), we can further simplify:
x = 2 * 29
x = 58
Therefore, the unknown side length, x, is equal to 58.
From the options provided the correct option is D.
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Markshisand
Which steps should be used to compare the fractions and
O Find a common numerator.
O Add the numerators and denominators to determine which sum is larger.
O Use the "greater than" symbol to show which denominator is larger.
O Multiply the first fraction by 2 and the second fraction by 3
Please awnser asap I will brainlist
Using simultaneous equation, the solution to the system of linear equations are 1223 $10 tickets, 1332 $20 tickets, and 763 $30 tickets were sold.
How many tickets of each kind has been sold?Let's solve the problem step by step.
Let:
x = number of $10 tickets sold
y = number of $20 tickets sold
z = number of $30 tickets sold
From the given information, we can form the following equations:
Equation 1: x + y + z = 3318 (Total number of tickets sold)
Equation 2: y = x + 109 (109 more $20 tickets than $10 tickets were sold)
Equation 3: 10x + 20y + 30z = 61760 (Total sales from ticket sales)
We can use these three equations to solve for the values of x, y, and z.
First, let's substitute Equation 2 into Equation 1:
x + (x + 109) + z = 3318
2x + 109 + z = 3318
2x + z = 3209 (Equation 4)
Now, let's substitute the value of y from Equation 2 into Equation 3:
10x + 20(x + 109) + 30z = 61760
10x + 20x + 2180 + 30z = 61760
30x + 30z = 59580
x + z = 1986 (Equation 5)
We now have a system of equations (Equations 4 and 5) with two variables (x and z). We can solve this system to find the values of x and z.
Multiplying Equation 4 by 30, and Equation 5 by 2, we get:
60x + 30z = 96270 (Equation 6)
2x + 2z = 3972 (Equation 7)
Now, subtract Equation 7 from Equation 6:
(60x + 30z) - (2x + 2z) = 96270 - 3972
58x + 28z = 92298
Simplifying, we have:
29x + 14z = 46149 (Equation 8)
Now, we can solve Equations 5 and 8 simultaneously:
x + z = 1986 (Equation 5)
29x + 14z = 46149 (Equation 8)
Multiplying Equation 5 by 14, and Equation 8 by 1, we get:
14x + 14z = 27804 (Equation 9)
29x + 14z = 46149 (Equation 8)
Now, subtract Equation 9 from Equation 8:
(29x + 14z) - (14x + 14z) = 46149 - 27804
15x = 18345
Divide both sides of the equation by 15:
x = 18345 / 15
x = 1223
Substituting the value of x into Equation 5, we can find z:
1223 + z = 1986
z = 1986 - 1223
z = 763
Now that we have the values of x and z, we can substitute them back into Equation 1 to find y:
1223 + y + 763 = 3318
y + 1986 = 3318
y = 3318 - 1986
y = 1332
Therefore, the solution to the problem is:
x = 1223 (number of $10 tickets sold)
y = 1332 (number of $20 tickets sold)
z = 763 (number of $30 tickets sold)
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Sample Answer: The activity type would be the input and the cost would be the output. Because there are two different costs for the same activity, this would not be a function.
Select any items on the list that you included in your response:
The activity type would be the input and the cost would be the output.
Because the rec center charges different prices for the same activity on different days, one input does not result in exactly one output.
No, the situation does not represent a function.
Due to the variation in costs for the same activity on different days, the situation does not represent a function.
Based on the information provided, it is clear that the situation does not represent a function. A function is a mathematical relationship where each input (or element from the domain) corresponds to exactly one output (or element from the range). In this case, the activity type serves as the input, and the cost represents the output.
However, since the rec center charges different prices for the same activity on different days, it violates the fundamental principle of a function, which states that one input cannot result in multiple outputs. The fact that there are two different costs associated with the same activity contradicts the definition of a function.
For example, let's consider the activity type "swimming." On Monday, the cost might be $10, but on Tuesday, the cost could be $15. This inconsistency demonstrates that one input, "swimming," leads to different outputs ($10 and $15). Therefore, it fails the criterion of a function.
To clarify, if each activity type had a consistent, unique cost associated with it, the relationship between the activity type and cost could be considered a function. However, since multiple costs exist for the same activity type, we cannot establish a definitive mapping between the inputs and outputs, rendering it non-functional.
In conclusion, due to the variation in costs for the same activity on different days, the situation does not represent a function.
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two dice are thrown simultaneously.find the probability of getting a) A sum less than 5
The probability of getting a sum less than 5 is 6/36, which can be simplified to 1/6.
When two dice are thrown simultaneously, the total number of outcomes is 36 (6 faces on each die, giving 6*6 = 36 possible outcomes).
To find the probability of getting a sum less than 5, we need to determine the favorable outcomes.
The possible favorable outcomes are:
1 + 1 = 2
1 + 2 = 3
1 + 3 = 4
2 + 1 = 3
2 + 2 = 4
3 + 1 = 4
There are a total of 6 favorable outcomes.
Thus, there is a 1 in 6 chance of getting a sum less than 5 when throwing two dice simultaneously.
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What is the area of this rectangle? Rectangle with width 5.1 cm and height 11.2 cm. Responses 16.3 cm2 16.3 cm, 2 32.6 cm2 32.6 cm, 2 57.12 cm2 57.12 cm, 2 571.2 cm2
The area of a rectangle with a width of 5.1 cm and a height of 11.2 cm is 57.12 cm².
To find the area of a rectangle, we multiply its length by its width. In this case, the width is given as 5.1 cm and the height (or length) is given as 11.2 cm.
Area = length × width
Area = 11.2 cm × 5.1 cm
Calculating the product, we get:
Area = 57.12 cm²
Therefore, the area of the rectangle is 57.12 cm².
The correct answer is: 57.12 cm².
It is important to note that when calculating the area of a rectangle, we should always include the appropriate unit of measurement (in this case, cm²) to indicate that we are dealing with a two-dimensional measurement. The area represents the amount of space covered by the rectangle's surface.
So, the area of a rectangle with a width of 5.1 cm and a height of 11.2 cm is 57.12 cm².
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A teaching hospital in South-West Part of Nigeria receives on the average 5 pregnant women with high blood pressure per week. What is the probability that on a particular week, the teaching hospital will receive:
1.) No high BP pregnant woman
Answer:
The probability that on a particular week, the hospital will receive on high BP pregnant woman is 0.0068
Step-by-step explanation:
We use the Exponential distribution,
Since we are given that on average, 5 pregnant women with high blood pressure come per week,
So, average = m = 5
Now, on average, 5 people come every week, so,
5 women per week,
so, we get 1 woman per (1/5)th week,
Hence, the mean is m = 1/5 for a woman arriving
and λ = 1/m = 5 = λ
we have to find the probability that it takes higher than a week for a high BP pregnant woman to arrive, i.e,
P(X>1) i.e. the probability that it takes more than a week for a high BP pregnant woman to show up,
Now,
P(X>1) = 1 - P(X<1),
Now, the probability density function is,
[tex]f(x) = \lambda e^{-\lambda x}[/tex]
And the cumulative distribution function (CDF) is,
[tex]CDF = 1 - e^{-\lambda x}[/tex]
Now, CDF gives the probability of an event occuring within a given time,
so, for 1 week, we have x = 1, and λ = 5, which gives,
P(X<1) = CDF,
so,
[tex]P(X < 1)=CDF = 1 - e^{-\lambda x}\\P(X < 1)=1-e^{-5(1)}\\P(X < 1)=1-e^{-5}\\P(X < 1) = 1 - 6.738*10^{-3}\\P(X < 1) = 0.9932\\And,\\P(X > 1) = 1 - 0.9932\\P(X > 1) = 6.8*10^{-3}\\P(X > 1) = 0.0068[/tex]
So, the probability that on a particular week, the hospital will receive on high BP pregnant woman is 0.0068
The table represents a linear relationship
X—2 0 4
Y-4 3 1
Which equation represents the table
Y=1/2x+5
y=-1/2x+3
Y=2x-3
Y=-4x+2
The linear relationship illustrated in the provided table can be effectively described by the equation Y = -4x + 2. Option D.
To determine the equation that represents the given table with the values of x and y, we can observe the pattern and find the equation of the line that fits these points.
Given the table:
X: 2 0 4
Y: -4 3 1
We can plot these points on a graph and see that they form a straight line.
Plotting the points (2, -4), (0, 3), and (4, 1), we can see that they lie on a line that has a negative slope.
Based on the given options, we can now evaluate each equation to see which one represents the line:
Y = 1/2x + 5
When we substitute the x-values from the table into this equation, we get the following corresponding y-values: -3, 5, and 6. These values do not match the given table, so this equation does not represent the table.
Y = -1/2x + 3
When we substitute the x-values from the table into this equation, we get the corresponding y-values: 4, 3, and 2. These values also do not match the given table, so this equation does not represent the table.
Y = 2x - 3
When we substitute the x-values from the table into this equation, we get the corresponding y-values: -4, -3, and 5. These values do not match the given table, so this equation does not represent the table.
Y = -4x + 2
When we substitute the x-values from the table into this equation, we get the corresponding y-values: -6, 2, and -14. Interestingly, these values match the y-values in the given table. Therefore, the equation Y = -4x + 2 represents the table.
In conclusion, the equation Y = -4x + 2 represents the linear relationship described by the given table. So Option D is correct.
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a) angle of line of From a point O in the school compound, Adeolu is 100 m away on a bearing N 35° E and Ibrahim is 80 m away on a bearing S 55° E. (a) How far apart are both boys? (b) (c) What is the bearing of Adeolu from point O, in three-figure bearings? What is the bearing of Ibrahim from point O, in three figure bearings? A boy walks 5 km due North and then 4 km due East. (a) Find the bearing of his current posi- tion from the starting point. (b) How far is the boy now from the start- ing point? A boy runs 200 m on a bearing of 230°.
a) Angle of line of sightFrom a point O in the school compound, Adeolu is 100 m away on a bearing N 35° E and Ibrahim is 80 m away on a bearing S 55° E. (a) How far apart are both boys? (b) (c) What is the bearing of Adeolu from point O, in three-figure bearings? What is the bearing of Ibrahim from point O, in three-figure bearings?The angle of the line of sight of Adeolu from the point O is given by:α = 90 - 35α = 55°.The angle of the line of sight of Ibrahim from the point O is given by:β = 90 - 55β = 35°.a) By using the Sine Rule, we can determine the distance between Adeolu and Ibrahim as follows:$
\frac{100}{sin55^{\circ}} = \frac{80}{sin35^{\circ}
100 sin 35° = 80 sin 55°=57.73 mT
herefore, both boys are 57.73 m apart. b) The bearing of Adeolu from the point O can be determined as follows:OAN is a right-angled triangle with α = 55° and OA = 100. Therefore, the sine function is used to determine the side opposite the angle in order to determine AN.
Thus:$$sin55^{\circ} = \frac{AN}{100}$$AN = 80.71 m.
To find the bearing, OAD is used as a reference angle. Since α = 55°, the bearing is 055°.
Therefore, the bearing of Adeolu from the point O is N55°E. c) Similarly, the bearing of Ibrahim from the point O can be determined as follows:OBS is a right-angled triangle with β = 35° and OB = 80. Therefore, the sine function is used to determine the side opposite the angle in order to determine BS.
Thus:$$sin35^{\circ} = \frac{BS}{80}$$BS = 46.40 m.
To find the bearing, OCD is used as a reference angle. Since β = 35°, the bearing is 035°.Therefore, the bearing of Ibrahim from the point O is S35°E. A boy walks 5 km due North and then 4 km due East. (a) Find the bearing of his current posi- tion from the starting point.
(b) How far is the boy now from the start- ing point?The boy's position is 5 km North and 4 km East from his starting position. The Pythagorean Theorem is used to determine the distance between the two points, which are joined to form a right-angled triangle. Thus
:$$c^2 = a^2 + b^2$$
where c is the hypotenuse, and a and b are the other two sides of the triangle. Therefore, the distance between the starting position and the boy's current position is:$$
c^2 = 5^2 + 4^2$$$$c^2 = 25 + 16$$$$c^2 = 41$$$$c = \sqrt{41} = 6.4 km$$
Therefore, the boy is 6.4 km from his starting point. (a) The bearing of the boy's current position from the starting point is given by the tangent function.
Thus:$$\tan{\theta} = \frac{opposite}{adjacent}$$$$\tan{\theta} = \frac{5}{4}$$$$\theta = \tan^{-1}{\left(\frac{5}{4}\right)}$$$$\theta = 51.34^{\circ}$$
Therefore, the bearing of the boy's current position from the starting point is N51°E.
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How are triangleABC and triangle ADE related? How do you know pls explain.
Triangle ABC and ADE are similar triangles
What are similar triangles?Similar triangles have the same corresponding angle measures and proportional side lengths.
This means that for two triangles to be similar, the corresponding angles must be equal and the ratio of corresponding sides of similar triangles are equal.
It has been shown that angles in ABC and ADE are equal.
To show that the ratio of corresponding sides are equal
6/12 = 8/16 = 10/20
The ratios all give a value of 1/2
Therefore we can say that the triangles ABC and ADE are similar.
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if a(x) = 3x+1 and b(x) = [tex]square root of x-4[/tex], what is the domain of (boa)(x)
The domain of (boa)(x) is [1, ∞].
What is a domain?In Mathematics and Geometry, a domain is the set of all real numbers (x-values) for which a particular equation or function is defined.
Based on the information provided above, we have the following functions:
a(x) = 3x+1
[tex]b(x) = \sqrt{x-4}[/tex]
Therefore, the composite function (boa)(x) is given by;
[tex]b(x) = \sqrt{3x+1 -4}\\\\b(x) = \sqrt{3x-3}[/tex]
By critically observing the graph shown in the image attached below, we can logically deduce the following domain:
Domain = [1, ∞] or {x|x ≥ 1}.
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What is the solution to this system of equations? 10 y = x - 2; y = - 0.5x + 7 10 (3, 2); (6, 4); (4, 6); (2, 3)
Answer:x=bx2 is the answer
Step-by-step explanation:
Mamá compra una caja de cubitos de azúcar. María se come la capa superior (S), que tiene 77 cubitos; después se come la cara lateral (L), que consta de 55 cubitos; y finalmente se come la cara frontal (F) también. ¿Cuántos cubitos quedan en la caja
Si mamá compra una caja de cubitos de azúcar y María se come la capa superior (S) de 77 cubitos, luego se come la cara lateral (L) de 55 cubitos y finalmente se come la cara frontal (F) también, podemos determinar cuántos cubitos quedan en la caja sumando todas las partes que María ha consumido.
En total, María se ha comido 77 cubitos de la capa superior, 55 cubitos de la cara lateral y otros cubitos de la cara frontal. Como no se proporciona el número exacto de cubitos de la cara frontal, no podemos calcular el número total de cubitos que quedan en la caja.
Para obtener la cantidad final de cubitos que quedan en la caja, necesitaríamos saber cuántos cubitos hay en la cara frontal y luego restar la suma de los cubitos que María se ha comido. Sin esa información adicional, no podemos determinar cuántos cubitos quedan en la caja.
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Factor using the GCMF.
6x4 + 12x³ + 15x²
2x^3 + 4x^2 + 5x
3(2x^4 +4x³+5x²)
3x² (2x² + 4x + 5)
3x^2
Answer:
3x² (2x² + 4x + 5)
Step-by-step explanation:
Step 1: Identify the coefficients.
In the given expression, the coefficients are 6, 12, and 15.
Step 2: Find the GCMF of the coefficients.
The GCMF is the largest number that can divide each coefficient evenly. In this case, the GCMF of 6, 12, and 15 is 3.
Step 3: Identify the variables.
The variables in the expression are x^4, x^3, and x^2.
Step 4: Find the GCMF of the variables.
The GCMF of the variables is the highest power of x that appears in each term. Here, it is x^2.
Step 5: Combine the GCMF of the coefficients and variables.
The GCMF of the coefficients (3) and the GCMF of the variables (x^2) can be multiplied together to get the overall GCMF: 3x^2.
Step 6: Factor out the GCMF from the expression.
To factor out the GCMF 3x^2, divide each term of the expression by 3x^2:
(6x^4 + 12x^3 + 15x^2) / (3x^2) = 2x^2 + 4x + 5
Step 7: Write the factored form.
The factored form of 6x^4 + 12x^3 + 15x^2 is 3x^2(2x^2 + 4x + 5).
100 Points! Geometry question. Photo attached. Please show as much work as possible. Thank you!
The value of x is equal to 15 units.
What is the triangle midpoint theorem?In Mathematics and Geometry, the Triangle midpoint theorem states that the line segment which joins the midpoints of two (2) sides of a triangle is parallel to the third side, and it's congruent to one-half of the third side.
In Mathematics and Geometry, a midsegment is a type of line segment that is used for connecting the midpoints of two sides of a triangle.
By applying the triangle midpoint theorem, we can determine the value of x as follows:
JH = 1/2(LM)
JH = 1/2 × 30
JH = x = 15 units.
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HELP ME PLEASE.
The figure below shows a rectangle ABCD having diagonals AC and DB:
Jimmy wrote the following proof to show that the diagonals of rectangle ABCD are congruent:
Jimmy's proof:
Statement 1: In triangle ADC and BCD, AD = BC (opposite sides of a rectangle are congruent).
Statement 2: Angle ADC = Angle BCD (angles of a rectangle are 90°
Statement 3:
Statement 4: Triangle ADC and BCD are congruent (by SAS postulate)
Statement 5: AC = BD (by CPCTC)
Which statement below completes Jimmy's proof? (1 point)
• AB=AB (reflexive property of equality)
• AB=AB (transitive property of equality)
O DC=DC (reflexive property of equality)
O DC=DC (transitive property of equality)
Statement 1: In triangle ADC and BCD, AD = BC (opposite sides of a rectangle are congruent).
Statement 2: Angle ADC = Angle BCD (angles of a rectangle are 90°).
Statement 3: DC = DC (reflexive property of equality).
Statement 4: Triangle ADC and BCD are congruent (by SAS postulate).
Statement 5: AC = BD (by CPCTC).
The statement that completes Jimmy's proof is:
DC = DC (reflexive property of equality)
The reflexive property of equality states that any quantity is equal to itself. In this case, statement 3 is stating that the diagonal DC is equal to itself, which is true by the reflexive property of equality.
Therefore, the completed proof is:
Statement 1: In triangle ADC and BCD, AD = BC (opposite sides of a rectangle are congruent).
Statement 2: Angle ADC = Angle BCD (angles of a rectangle are 90°).
Statement 3: DC = DC (reflexive property of equality).
Statement 4: Triangle ADC and BCD are congruent (by SAS postulate).
Statement 5: AC = BD (by CPCTC).
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35
The cost of packing a box of chocolates is given by x2, where x is the number of chocolates (a box can never have fewer than 3 chocolates). If the
weight of a box of chocolates is given by x + 2, what is the cost of packaging per weight unit?
OA.
B.
O c.
OD.
O
++2
2² +1
²+1
--
Reset
Next
The cost of packaging per weight unit is x + 1.
To derive this answer, we first need to understand the given information. The cost of packing a box of chocolates is given by [tex]x^2[/tex], where x is the number of chocolates. However, we also know that a box can never have fewer than 3 chocolates.
Now, let's calculate the weight of a box of chocolates. It is given by x + 2.
To find the cost of packaging per weight unit, we need to divide the cost of packing by the weight of the box. Therefore, the cost of packaging per weight unit can be calculated as ([tex]x^2[/tex]) / (x + 2).
Simplifying this expression, we can rewrite it as ([tex]x^2[/tex]) / (x + 2) = (x + 1) - 1 / (x + 2).
Hence, the cost of packaging per weight unit is x + 1, which means that for every unit of weight, the cost of packaging is x + 1.
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Using substitution method, the solution to the system of linear equations
x = 26667 and y = 533
What is the solution to the system of linear equation?Let's set up the system of equations based on the given information.
Let:
x = number of currency A
y = number of currency B
Equation 1: x + y = 3200 (Total budgeted spending money)
Equation 2: 1.30x + 1.50y = Total cost of purchasing currency (to be determined)
Based on the information provided, Abigail wants to have five times as much currency A as currency B:
Equation 3: x = 5y
Now, let's solve the system of equations using substitution method
Substitute Equation 3 into Equation 1:
5y + y = 3200
6y = 3200
y = 3200 / 6
y = 533.33 (approximately)
Substitute the value of y back into Equation 3 to find x:
x = 5(533.33)
x = 2666.67 (approximately)
Now, let's calculate the total cost of purchasing currency using Equation 2:
Total cost of purchasing currency = 1.30x + 1.50y
Total cost of purchasing currency = 1.30(2666.67) + 1.50(533.33)
Total cost of purchasing currency = 3466.67 + 800
Total cost of purchasing currency = 4266.67 (approximately)
Therefore, the system of equations has a solution where:
x ≈ 2667 (number of currency A)
y ≈ 533 (number of currency B)
This means that Abigail should exchange approximately 2667 units of currency A and 533 units of currency B. The total cost of purchasing the currency would be approximately $4266.67.
In practical terms, this means that Abigail will have a budgeted spending money of $3200, and she plans to allocate it between currency A and currency B based on the given exchange rates and her preference of having five times as much currency A as currency B.
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21 20 15 22 19 23 17 what is the median and range
The median would be 22:
Step-by-step explanation: The median is the number that is in the middle
Simply defined, the median is the number right in the middle of a set.
But before I get down to finding the median, I will order the numbers from least to greatest:
15, 17, 19, 20, 21, 22, 23
Now, the middle number is 20. So that's the median.
As for the range, it's the difference between the largest number and the smallest one:
Range = Greatest number - Smallest number
= 23 - 15
= 6
In summary, the median, the number in the middle, is 20, and the range, the difference between the largest number and the smallest one, is 6.
16
Find X.
25
X
Pythagorean Theorem
Answer:
3√41
Step-by-step explanation:
a^2+b^2=c^2
rearrange this to make x (a or b it doesn't matter) the subject
√b=c^2-a^2
now substitute this in:
√b=25^2-16^2
=369
then simple do the square root
√369
=3√41
Hope this helps!
he table displays the total cost, y, of purchasing x tickets for the carnival.
A 2-column table with 3 rows. Column 1 is labeled Tickets, x with entries 11, 12, 13. Column 2 is labeled Total Cost, y (dollars) with entries 27.50, 30.00, 32.50.
Which conclusions can you draw from the data shown in the table? Select all that apply.
Twelve tickets cost $30.00.
Thirty tickets cost $12.00.
Each additional ticket costs $2.50.
The table is a partial representation.
(27.50, 11), (30, 12) and (32.50, 13) are the ordered pairs represented in the table.
The conclusions that can be drawn from the data shown in the table are:
- Twelve tickets cost $30.00.
- Each additional ticket costs $2.50.
- The table is a partial representation.
- (27.50, 11), (30, 12), and (32.50, 13) are the ordered pairs represented in the table.
The statement "(27.50, 11), (30, 12), and (32.50, 13) are the ordered pairs represented in the table" is correct. From the data shown in the table, we can draw the following conclusions:
1. Twelve tickets cost $30.00: Looking at the "Tickets" column, we can see that the entry "12" corresponds to the "Total Cost" entry of $30.00. Therefore, we can conclude that purchasing twelve tickets would cost $30.00.
2. Each additional ticket costs $2.50: By examining the "Tickets" column, we can observe that for each increase of one ticket (from 11 to 12, and from 12 to 13), the "Total Cost" in the second column increases by $2.50. This consistent pattern suggests that each additional ticket costs $2.50.
3. The table is a partial representation: The table only displays three rows of data, showing the "Tickets" and "Total Cost" for x values of 11, 12, and 13. Since the table does not provide information for all possible values of x, it is a partial representation of the relationship between the number of tickets and the total cost.
The statement "Thirty tickets cost $12.00" is not supported by the given data. The table does not include an entry for 30 tickets, and none of the given entries correspond to that value.
These ordered pairs match the values shown in the table, with the first element representing the number of tickets (x) and the second element representing the total cost (y) in dollars.
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Let f(x)
f'(x) =
-
=
1
v2x2+5x+3
Question Help:
(Use sqrt(N) to write √N)
Video Written Example
The derivative of the function [tex]f(x) = \frac{1}{\sqrt{2x^2 + 5x + 3}}[/tex] is [tex]f'(x) = -\frac{4x + 5}{2((x + 1)(2x + 3))^\frac{3}{2}}[/tex]
How to calculate the derivative of the functionFrom the question, we have the following parameters that can be used in our computation:
[tex]f(x) = \frac{1}{\sqrt{2x^2 + 5x + 3}}[/tex]
Factor the expression
So, we ave
[tex]f(x) = \frac{1}{\sqrt{(x + 1)(2x + 3)}}[/tex]
The derivative of the function can be calculated using as follows:
[tex]f'(x) = (-\frac{1}{2})((x + 1)(2x + 3))^{-\frac{1}{2} - 1} \cdot \frac{d}{dx}[(x + 1)(2x + 3)][/tex]
Next, we have
[tex]f'(x) = -\frac{\frac{d}{dx}(x + 1) \cdot (2x + 3) + (x + 1) \cdot \frac{d}{dx}(2x + 3)}{2((x + 1)(2x + 3))^\frac{3}{2}}[/tex]
Differentiate
[tex]f'(x) = -\frac{1 \cdot (2x + 3) + (x + 1) \cdot 2}{2((x + 1)(2x + 3))^\frac{3}{2}}[/tex]
This gives
[tex]f'(x) = -\frac{2x + 3 + 2x + 2}{2((x + 1)(2x + 3))^\frac{3}{2}}[/tex]
Evaluate the like terms
[tex]f'(x) = -\frac{4x + 5}{2((x + 1)(2x + 3))^\frac{3}{2}}[/tex]
Hence, the derivative of the function is [tex]f'(x) = -\frac{4x + 5}{2((x + 1)(2x + 3))^\frac{3}{2}}[/tex]
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Find each value given the following function:
Answer:
Step-by-step explanation:
1) f(-4) --> if x < or equal to 3
2) 1/(-4)-4
3) The answer is - 1/8
Sample Response/Explanation: Let x represent the number of tickets sold, and y represent the total amount of money raised. Since each ticket is $2.50, the total amount of money raised is equal to $2.50 times the number of tickets. The equation would be y = 2.50x. Select each of the following that you included in your response. The x variable represents the number of tickets sold. The y variable represents the total amount of money raised from ticket sales. The equation for the scenario is y = 2.50x.
The x variable represents the number of tickets sold.
The y variable represents the total amount of money raised from ticket sales.
The equation for the scenario is y = 2.50x.
In the given scenario, the number of tickets sold is represented by the variable x, and the total amount of money raised from ticket sales is represented by the variable y. Since each ticket is priced at $2.50, the equation relating the number of tickets sold (x) and the total amount of money raised (y) is y = 2.50x.
Here's an explanation of each element in the response:
1. The x variable represents the number of tickets sold: This statement correctly identifies the variable x as representing the number of tickets sold. In the equation y = 2.50x, x represents the independent variable, which is the quantity we want to determine.
2. The y variable represents the total amount of money raised from ticket sales: This statement correctly identifies the variable y as representing the total amount of money raised from ticket sales. In the equation y = 2.50x, y represents the dependent variable, which is determined based on the value of x.
3. The equation for the scenario is y = 2.50x: This equation is derived from the given information that each ticket is priced at $2.50. Multiplying the price per ticket by the number of tickets sold gives the total amount of money raised, which is represented by y in the equation.
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The slope of a parabola
�
=
3
�
2
−
11
�
+
10
y=3x
2
−11x+10 at a point
�
P is 7. Find the
�
−
y− coordinate of the point
�
P
The y-coordinate of point P is 1.To find the y-coordinate of the point P on the parabola y = 3x^2 - 11x + 10 where the slope is 7, we can differentiate the equation to find the derivative. The derivative of y = 3x^2 - 11x + 10 is y' = 6x - 11.
To find the x-coordinate of point P, we can set the derivative equal to the given slope: 6x - 11 = 7. Solving for x, we get x = 3.
To find the y-coordinate of point P, we substitute the x-coordinate back into the original equation: y = 3(3^2) - 11(3) + 10. Simplifying, we find y = 1.
Therefore, the y-coordinate of point P is 1.
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A university is interested in determining the average statistics anxiety score for all undergraduate students in the U.S. For a random sample of 33 undergraduate students, it is found that the average average statistics anxiety score is 39.4 with a standard deviation of 0.9. Assume that the statistics anxiety scores for all undergraduate students in the U.S is normally distributed. A 98% confidence interval for the true mean statistics anxiety score μ is closest to.
The 98% confidence interval for the true mean statistics anxiety score (μ) is approximately (39.037, 39.763).
To calculate the 98% confidence interval for the true mean statistics anxiety score (μ) for all undergraduate students in the U.S., we can use the formula:
Confidence interval = sample mean ± (critical value * standard error)
First, we need to find the critical value associated with a 98% confidence level. Since we are assuming a normal distribution, we can use the Z-table or a statistical software to find this value. For a 98% confidence level, the critical value is approximately 2.33.
Next, we calculate the standard error (SE) using the formula:
SE = standard deviation / √sample size
In this case, the standard deviation is 0.9 and the sample size is 33. Plugging these values into the formula, we get: SE = 0.9 / √33 ≈ 0.156
Now, we can calculate the confidence interval:Confidence interval = 39.4 ± (2.33 * 0.156)
Simplifying the expression:Confidence interval ≈ 39.4 ± 0.363
This gives us the interval (39.037, 39.763). This means we are 98% confident that the true mean statistics anxiety score for all undergraduate students in the U.S. falls within this interval.
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A deposit of $2,580 is placed into a retirement fund at the beginning of every 6 months for 15 years. The fund earns 3% annual interest, compounded biannually and paid at the end of the 6 months. How much is in the account right after the last deposit?
Round your answer to the nearest dollar.
The amount in the account right after the last deposit, rounded to the nearest dollar, is approximately $3,982.
To calculate the amount in the account right after the last deposit, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal (initial deposit)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case, the principal is $2,580, the annual interest rate is 3% (0.03 in decimal form), the interest is compounded biannually (twice per year), and the total duration is 15 years.
First, let's calculate the number of compounding periods:
Since interest is compounded twice a year, and there are 15 years, the total number of compounding periods is 15 * 2 = 30.
Now, we can plug the values into the formula:
A = $2,580(1 + 0.03/2)^(2*30)
A = $2,580(1 + 0.015)^60
A = $2,580(1.015)^60
Using a calculator, we find that (1.015)^60 ≈ 1.545.
A = $2,580 * 1.545
A ≈ $3,982.10
Therefore, the amount in the account right after the last deposit, rounded to the nearest dollar, is approximately $3,982.
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X
45°
X =
Find x.
17
X
45°
AVC
The value of the length labelled x is 17√2/2
How to determine the valueTo determine the value of the variable, we need to know the six different trigonometric identities.
These identities are listed as;
cosinesinetangentcotangentsecantcosecantFrom the information given, we have that;
Using the sine identity
sin theta = opposite/hypotenuse
But we have from the image that;
Theta = 45 degrees
Opposite side of the triangle = x
Hypotenuse side = 17
Now, substitute the values, we get;
sin 45 = x/17
find the value, we have;
1/√2 = x/17
cross multiply and rationalize
x = 17√2/2
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Use the chain rule to find the derivative of
f(x) = 4√/8x³ + 2x4
Type your answer without fractional or negative exponents. Use sqrt(x) for √√x.
X.
f'(x) =
Using chain rule, the derivative of f(x) = 4√(8x³ + 2x⁴) is
f'(x) = (2/√(8x³ + 2x⁴)) * (24x² + 8x³)
What is the derivative of the function?To find the derivative of the function f(x) = 4√(8x³ + 2x⁴), we can use the chain rule.
Let's break down the function into its components:
u(x) = 8x³ + 2x⁴ (inside function)
v(u) = 4√u (outer function)
To find the derivative, we apply the chain rule, which states:
(f(g(x)))' = f'(g(x)) * g'(x)
In this case, f(g(x)) = v(u(x)), and g(x) = u(x).
First, let's find the derivative of the inner function u(x):
u'(x) = 24x² + 8x³ (using the power rule for differentiation)
Next, let's find the derivative of the outer function v(u):
v'(u) = 4 * (1/2) * 1/√u = 1/√2u = 2/√u
Now, we can apply the chain rule:
f'(x) = v'(u(x)) * u'(x)
f'(x) = (2/√u) * (24x² + 8x³)
f'(x) = (2/√(8x³ + 2x⁴)) * (24x² + 8x³)
The derivative of the function is (2/√(8x³ + 2x⁴)) * (24x² + 8x³)
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For the following equations find 1) the general form for all solutions 2) the solutions on the interval (pi/2)<=θ<3pi in terms of pi.
cscθ=2
For the following equations:
1) In general, the solutions can be expressed as θ = π/6 + 2πn or θ = 5π/6 + 2πn, where n is an integer.
2) The solutions within the interval (π/2) ≤ θ < 3π can be represented as θ = 7π/6 or θ = 11π/6, both in terms of π.
To solve the equation cscθ = 2, we need to find the values of θ that satisfy the equation.
1) General Form for All Solutions:
The reciprocal of sine is cosecant (csc), so we can rewrite the equation as 1/sinθ = 2. To solve for θ, we can take the reciprocal of both sides:
sinθ = 1/2
Now, we need to determine the values of θ where the sine function equals 1/2. The sine function is positive in the first and second quadrants, so we'll focus on those quadrants.
In the first quadrant (0 ≤ θ < π), the reference angle with a sine of 1/2 is π/6.
In the second quadrant (π < θ < 2π), the reference angle with a sine of 1/2 is also π/6.
To account for all solutions, we can add multiples of the period of sine (2π) to the reference angles. Therefore, the general form for all solutions is:
θ = π/6 + 2πn or θ = 5π/6 + 2πn
where n is an integer representing the number of periods of sine added.
2) Solutions on the Interval (π/2) ≤ θ < 3π in Terms of π:
For the given interval, we need to find the values of θ that satisfy the equation and lie within the interval (π/2) ≤ θ < 3π.
From the general form, we can see that the solutions that satisfy the interval are:
θ = π/6 + 2π or θ = 5π/6 + 2π
Simplifying these expressions gives us:
θ = 7π/6 or θ = 11π/6
Therefore, the solutions on the interval (π/2) ≤ θ < 3π in terms of π are:
θ = 7π/6π or θ = 11π/6π.
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Heather is making chocolate biscuits
she has:
2kg of flour
1kg of butter
340g of icing sugar
200g of chocolate
here is the list of ingredients for making 20 biscuits.
100g of flour
120g of butter
80g of icing sugar
25g of chocolate
work out how mnany bisuits she can make
Therefore, Heather can make a maximum of 4 biscuits with the given ingredient quantities.
To determine how many biscuits Heather can make, we need to compare the amount of each ingredient required for a single biscuit to the total amount of ingredients she has.
Let's calculate the number of biscuits she can make based on the ingredient quantities provided:
First, we need to find the ratio of each ingredient required per biscuit:
Flour: 100g per biscuit
Butter: 120g per biscuit
Icing sugar: 80g per biscuit
Chocolate: 25g per biscuit
Next, we divide the total amount of each ingredient by the respective ratio to find the maximum number of biscuits she can make:
Flour: 2kg / 100g = 20 biscuits
Butter: 1kg / 120g = 8.33 biscuits (approximately)
Icing sugar: 340g / 80g = 4.25 biscuits (approximately)
Chocolate: 200g / 25g = 8 biscuits
Since we can only make a whole number of biscuits, the limiting factor is the icing sugar. Heather can only make 4 biscuits since she has 4.25 times the required amount of icing sugar.
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