Given AB and AC are lines that are tangent to the circle with
the measure of angle BAC = 40°, what is the measure of angle BDC?
Answers
Given AB and AC are lines that are tangent to the circle with the measure of angle BAC = 40°, ∠BDC is 140°.
A tangent to a circle is a line that intersects the circle at a single point. The point at which the tangent intersects the circle is known as the point of tangency. The tangent is perpendicular to the circle's radius, with which it meets.
You've been handed two tangent lines. You will also be handed a four-sided figure. All four-sided figures have 360 degrees of rotation. At 90 degrees, a radius meets a tangent.
∠BDA = 90°
∠DCA = 90°
∠BCA = 40°
All the angles in total make 360°, so:
∠BDA + ∠DCA + ∠BCA + ∠BDC = 360
90 + 90 + 40 + ∠BDC = 360
220 + ∠BDC = 360
∠BDC = 360 - 220
= 140°
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Correct question:
Given AB and AC are lines that are tangent to the circle with the measure of angle BAC = 40°, what is the measure of angle BDC? Image is attached below.
Related Questions
Use tha appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. $26,000 invested at 3.65% annual interest
for 2 years compounded
(a) daily (n = 365); (b) continuously
Answers
Amount that will be in the account after 2 years with daily compounding is $28,484.03.
Amount that will be in the account after 2 years with continuous compounding is $28,498.84.
What is appropriate proceedure to calculate annual interest?The appropriate compound interest formula is:
[tex]A = P(1 + r/n)^{nt[/tex]
A is the amount.
P is the principal.
r is the annual interest rate.
n is the number of times the interest is compounded all year.
t is the number of years.
(a) For daily compounding (n = 365), we have:
A = 26000(1 + 0.0365/365)³⁶⁵*²
A = 26000(1 + 0.0001)⁷³⁰
A = 26000(1.0001)⁷³⁰
A = 28,484.03
Therefore, the amount that will be in the account after 2 years with daily compounding is $28,484.03.
(b) For continuous compounding, we have:
A = P[tex]e^{rt[/tex]
e is the mathematical constant almost equal to 2.71828.
A = 26000[tex]e^{0.0365*2[/tex]
A = 26000[tex]e^{0.073[/tex]
A = 28,498.84
Therefore, the amount that will be in the account after 2 years with continuous compounding is $28,498.84.
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Your friend makes a stem-and-leaf plot of the data. 51, 25, 47, 42, 55, 26, 50, 44, 55 Student work is shown. A stem and leaf plot. A vertical line separates each stem from its first leaf. The first row has a stem of 2 and leaves 5 and 6. The second row has a stem of 4 and leaves 2, 4, and 7. The third row has a stem of 5 and leaves 0, 1, 5, and 5. The key shows 4 vertical bar 2 is equal to 42. Is your friend correct? Responses yes yes no no Question 2 Explain your reasoning.
Answers
Yes, your friend is not correct about the stem and leaf plot.
How to design the stem and leaf plot ?The stem and leaf plot made by your friend is:
Stem | Leaves
2 | 5, 6
4 | 2, 4, 7
5 | 0, 1, 5, 5
Key : 4 | 2 = 42
When the data points from these are taken, we have :
25 , 26 , 42 , 44 , 47 , 50, 51, 55, 55
This is the same as the data provided of :
51, 25, 47, 42, 55, 26, 50, 44, 55
So, your friend's stem and leaf plot is indeed correct.
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Drake was trying to write an equation to help him predict the cost of his monthly phone bill.
He is charged $35 just for having a phone, and his only additional expense comes from the number of text that he sends,
He is charged $0. 05 for each text. Name the function C(t), where C(t) is the cost of bill according to number of text.
Answers
The function for Drake's monthly phone bill, C(t), is C(t) = 35 + 0.05t, where t represents the number of texts sent.
Drake has two components to his monthly phone bill: the base cost of $35 and the additional cost for texts sent. The base cost is fixed, meaning it does not change, so we can represent it as a constant value in the function (35).
The cost per text is $0.05, which means it varies depending on the number of texts sent (t). To find the total cost (C(t)), we need to add the fixed cost ($35) and the variable cost ($0.05 times the number of texts). Therefore, the function representing Drake's monthly phone bill cost is C(t) = 35 + 0.05t.
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Susan bought two gifts. One package is a rectangular prism with a base length of 4 inches, a base width of 2 inches, and a height of 10 inches. The other package is a cube with a side length of 5 inches. Which package requires more wrapping paper to cover? What is the total amount of wrapping paper Susan must use to cover both packages? You must show your work to earn full credit
Answers
The package that requires more wrapping paper to cover is the cube. The total amount of wrapping paper Susan must use to cover both packages is 286 square inches.
Let's find the surface area of both packages to determine which requires more wrapping paper and the total amount needed.
1. Rectangular prism:
Surface area = 2lw + 2lh + 2wh
where l = length, w = width, h = height
Surface area = 2(4)(2) + 2(4)(10) + 2(2)(10)
Surface area = 16 + 80 + 40 = 136 square inches
2. Cube:
Surface area = 6s²
where s = side length
Surface area = 6(5)² = 6(25) = 150 square inches
The cube requires more wrapping paper to cover as its surface area is 150 square inches, compared to the rectangular prism's 136 square inches. The total amount of wrapping paper Susan must use for both packages is 136 + 150 = 286 square inches.
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My sister is 16 years old. My brother says that his age minus nighteen is equal to my sister's age. How old is my brother? Write a equation with b as variable
Answers
The brother is 35 years old.
How to use the variable "b" to represent the brother's age?Let's use the variable "b" to represent your brother's age.
According to the problem, your brother's age minus nineteen (b - 19) is equal to your sister's age, which is 16. So we can write an equation:
b - 19 = 16
To solve for b, we can add 19 to both sides of the equation:
b - 19 + 19 = 16 + 19
Simplifying, we get:
b = 35
Therefore, your brother is 35 years old.
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2. A stone with a speed of 0.80 m/s rolls off the edge of a table 1.5 m high.
a. How long does it take to hit the floor
b. How far from the table will it hit floor
Answers
Answer:
Step-by-step explanation:
a. To find the time it takes for the stone to hit the floor, we can use the formula t = sqrt(2h/g), where h is the height of the table and g is the acceleration due to gravity. Plugging in the values, we get:
t = sqrt(2(1.5 m)/9.8 m/s^2) = 0.55 seconds.
b. To find the horizontal distance traveled by the stone, we can use the formula d = vt, where v is the initial velocity of the stone and t is the time it takes to hit the floor. Plugging in the values, we get:
d = (0.80 m/s) * (0.55 s) = 0.44 meters.
Therefore, the stone will hit the floor after 0.55 seconds and will travel 0.44 meters from the table.
Of the following, which option or options would help make this graph less misleading? i. the scale on the x-axis should be resized. ii. the scale on the y-axis should be resized. iii. the identity of the two parks should be more clearly differentiated. a. i and ii b. ii and iii c. iii only d. i and iii
Answers
The option or options that would help make the graph less misleading are:
d. i and iii. The scale on the x-axis should be resized. The identity of the two parks should be more clearly differentiated.
Resizing the scale on the x-axis (option i) would help provide a clearer picture of the difference between the two parks, as it would make it easier to see the differences in the number of visitors between the two parks.
Differentiating the identity of the two parks more clearly (option iii) would also help reduce confusion and provide a more accurate representation of the data.
Resizing the scale on the y-axis (option ii) may not be necessary in this case, as the existing scale is appropriate and accurately represents the data.
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Question 2. Enter the correct answer in the box.
Answers
The given equation, a = v²/r, solved for r is:
r = v²/a
Subject of formulae: Solving the equation for rFrom the question, we are to solve the given equation for r
From the given information,
The given equation is
a = v²/r
To solve the equation for r means we should isolate the variable r
Solving the equation for r
a = v²/r
Multiply both sides of the equation by r
a × r = v²/r × r
ar = v²
Divide both sides of the equation by a
ar/a = v²/a
r = v²/a
Hence, the equation solved for r is:
r = v²/a
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Which region contains viable solutions to the systems of inequalities within the context of the situation? The situation is Everett wanted to create chocolate milk using whole milk and chocolate syrup. He wanted his chocolate milk to have less than 8 grams (g) of fat and less than 28 grams (g) of sugar. He drew the graph shown where x is the amount of whole milk in ounces (oz) and y is the amount of chocolate syrup in ounces (oz).
Answers
The region that contains viable solutions to the systems of inequalities is region H
Which region contains viable solutions to the systems of inequalities?From the question, we have the following parameters that can be used in our computation:
Chocolate milk to have less than 8 grams (g) of fat Also less than 28 grams (g) of sugar.This means that the region viable solutions to the systems of inequalities is the region below the inequallity line
In this case, the region is the region G
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OAB is a triangle.
O A = a OB = b
C is the midpoint of OA.
D is the point on AB such that AD: DB = 3:1
E is the point such that OB = 2BE
Using a vector method, prove that the points C, D and E lie on the same straight
line.
Input note: express CE in terms of CD
(5 marks)
â
Answers
This evaluated expression is a scalar multiple of -4, which projects that vectors CD and CE are collinear. Then, points C, D, and E lie on the same straight line.
Let us proceed by evaluating the vector CD. Then C is the midpoint of OA, we can evaluate the vector CD by subtracting vector CO from vector OD.
Vector CO = 1/2 × Vector OA
= 1/2 × (a + b)
= 1/2a + 1/2b
Vector OD = 3/4 × Vector AD
= 3/4 × (3/4a - 1/4b)
= 9/16a - 3/16b
Vector CD = Vector OD - Vector CO
= (9/16a - 3/16b) - (1/2a + 1/2b) = 5/16a - 5/16b
Then the value of the vector CE is
OB = 2BE,
we can evaluate the vector BE by dividing vector OB by 2.
Vector BE = 1/2 × Vector OB
= 1/2 × b
= 1/2b
Vector CE = Vector CO + Vector OE
Vector OE = Vector OB - Vector OE
= b - Vector BE
= b - 1/2b
= 1/2b
Vector CE = Vector CO + Vector OE
= (1/2a + 1/2b) + (1/2b)
= 1/2a + b
Then we have to show that vectors CD and CE are collinear. Two vectors are collinear if one is a scalar multiple.
CE can be expressed in terms of CD
CE / CD
= ((1/2a + b) / (5/16a - 5/16b))
Applying simplification for this expression
CE / CD
= (-8a - 8b) / (5a - 5b)
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can someone help me answer #17 using square roots?
Answers
Answer: x = ± [tex]\frac{1}{2}[/tex]
Step-by-step explanation:
Equation:
4x² + 10 = 11 > bring everything over to other side
> first subtract 10 from both sides
4x² = 1 > Divide by 4 on both sides
x² = [tex]\frac{1}{4}[/tex] >Take square root of both sides
> When you take square root there is a ±
x = ± [tex]\sqrt{(\frac{1}{4} )}[/tex] > take the square root of both top and bottom
x = ± [tex]\frac{1}{2}[/tex]
2. Iwo functions t and g are definea on the set R of real numbers by f: x x² - 2x - 4. g: x → X - 1 Find the value fx for which f(x) = (x) = m - 4.
Answers
Answer:
We are given that:
- f(x) = x² - 2x - 4
- g(x) = x - 1We want to find fx for which f(x) = g(x) - 4, or in other words:
- f(x) = x - 1 - 4
- f(x) = x - 5
We can solve forTo find the value of x for which f(x) = g(x) - 4 (which is what I assume you meant by "f(x) = (x) = m - 4"), we can set up the following equation:
f(x) = g(x) - 4
Substituting the given expressions for f(x) and g(x), we get:
x² - 2x - 4 = x - 1 - 4
Simplifying, we have:
x² - 3x - 3 = 0
We can solve for x using the quadratic formula:
x = (-(-3) ± sqrt((-3)² - 4(1)(-3))) / (2(1))
x = (3 ± sqrt(21)) / 2
Therefore, the two values of x for which f(x) = g(x) - 4 are:
- x = (3 + sqrt(21)) / 2
- x = (3 - sqrt(21)) / 2
Solve for x, t, r and round to the nearest hundredth
Answers
Answer:
x = 14°
t = 12.367 ~ 12.4
r = 2.999 ~ 3
Step-by-step explanation:
1st we can find x by sum theory which is the sum of all side equal to 180° .
x + 90° + 76° = 180 °
x + 166° = 180°
x= 180° - 166°
x = 14° ... So the unknown angle is 14°
and we also can solve hypotenus t and adjecent r by using sin amd cos respectively by angle 76° .
sin(76) = 12/t
sin(76) t = 12 ....... criss cross it
t = 12 / sin(76) ....... divided both side by sin(76)
t = 12.367 ~ 12.4 ....... result
And
cos(76) = r / 12.4
r = cos(76) × 12.4 .......criss cross
r = 2.999 ~ 3 ....... amswer and i approximate it
Find the length of the following parametric curve. x = 6 + it, y = 4 + 43/2, 03752. 0 << Enter your answer symbolically, as in these examples
Answers
To find the length of the parametric curve, we can use the formula:
L = ∫a^b √(dx/dt)² + (dy/dt)² dt
Substituting the given values, we get:
L = ∫0^1 √(6)² + (43/2, 03752)² dt
Simplifying:
L = ∫0^1 √(36 + (43/2, 03752)²) dt
Therefore, the length of the parametric curve is:
L = √(36 + (43/2, 03752)²)
In mathematics, substitution refers to the process of replacing a variable or expression in an equation or formula with another variable or expression that has the same value.
For example, if we have an equation: 2x + 3y = 7, and we want to substitute x with the value 4, we replace x with 4 to get:
2(4) + 3y = 7
8 + 3y = 7
We can then solve for y to find its value.
Substitution is a commonly used technique in algebra and calculus to simplify expressions, solve equations and evaluate functions.
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Sean poured 2160 cm cubed of lemonade into some containers which
were 9 cm long, 8cm wide, and 6 cm high. Each container was completely
filled with lemonade. How many containers were there? There were
containers. *
Answers
The number of cubical containers which are 9 cm long, 8cm wide, and 6 cm high completely filled with lemonade is 5.
volume of lemonade = 2160 cm³
Dimensions of container
L = 9 cm , B = 8 cm , H = 6 cm
Volume of container = L× B × H
Volume of container = 9×8×6
Volume of container = 432 cm³
To find the number of cubical containers filled we use
Number of containers filled = volume of lemonade/volume of the container
putting the value in formula
Number of container filled = 2160/432
Number of container filled = 5
Total number of container filled with lemonade is 5
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A cone has a volume of 1230. 88 units cubed and a diameter of 14 units. How many units is the height of the cone? Use 3. 14 for pi
Answers
Answer: 24 unit
Step-by-step explanation:
Write the equation below in standard and factored form y= -(x-1)^2+25
Answers
According to the national oceanic and atmospheric administration (noaa), between 1851 and 2013 there were 290 hurricanes that hit the u.s. coast. of these, 117 were category 1 hurricanes, 76 were category 2 hurricanes, 76 were category 3 hurricanes, 18 were category 4 hurricanes, and 3 were category 5 hurricanes. make a probability distribution for this data. if a hurricane hits the u.s. coast, what is the probability that the hurricane will be a category 1 hurricane?
Answers
The probability of a hurricane hitting the U.S. coast and being a category 1 hurricane is 0.403, or 40.3%.
The National Oceanic and Atmospheric Administration (NOAA) is a federal agency that is responsible for monitoring and predicting changes in the Earth's environment, including the atmosphere and oceans. One of their main responsibilities is to track and study hurricanes that affect the United States.
According to their data, there have been a total of 290 hurricanes that have hit the U.S. coast between 1851 and 2013. These hurricanes are categorized based on their wind speeds, with categories ranging from 1 to 5.
To create a probability distribution for this data, we need to calculate the probability of each category of hurricane occurring. We can do this by dividing the number of hurricanes in each category by the total number of hurricanes.
Category 1 hurricanes: 117/290 = 0.403
Category 2 hurricanes: 76/290 = 0.262
Category 3 hurricanes: 76/290 = 0.262
Category 4 hurricanes: 18/290 = 0.062
Category 5 hurricanes: 3/290 = 0.010
Therefore, the probability of a hurricane hitting the U.S. coast and being a category 1 hurricane is 0.403, or 40.3%. This means that out of all the hurricanes that have hit the U.S. coast, about 40% of them were category 1 hurricanes. It is important to note that this probability distribution is based on historical data and may not accurately predict the likelihood of future hurricanes.
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Can someone help and explain the terms and sequence with answers please?
29, 22, 15, 8
B) these are the first five terms of another sequence.
4, 7, 12, 19, 28
Find the nth term.
Answers
B) The given sequence is increasing by successive odd integers, starting with 3. To find the nth term of this sequence, we can use the formula:
nth term = first term + (n-1) * common difference
The first term of this sequence is 4, and the common difference is the successive odd integers. The difference between each term and the previous term is 3, 5, 7, 9, and so on. So, the common difference is 2 more than the index of the term.
For example, the second term (n=2) is 7, which is 3 more than the first term. Therefore, the common difference is 3. The third term (n=3) is 12, which is 5 more than the second term. Therefore, the common difference is 5-3=2.
Thus, we can write the nth term as:
nth term = 4 + (n-1) * (n+1)/2
For example, the fourth term (n=4) would be:
nth term = 4 + (4-1) * 5/2 = 4 + 3*5/2 = 4 + 15/2 = 11.5
Therefore, the nth term of the sequence is 4 + (n-1) * (n+1)/2.
Write a function rule for the statement.
the output is eight less than the input
Answers
The function rule for the statement "the output is eight less than the input" is a simple mathematical expression that represents a relationship between the input and output values.
In this case, it can be expressed as Output = Input - 8. The function takes the input value, subtracts 8 from it, and returns the result as the output value. This rule ensures that the output will always be eight units smaller than the input. For example, if the input is 15, the output will be 7. This function rule can be used to perform calculations or model various scenarios where the output is consistently eight units less than the input.
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Use the formula SA = 2π
rh + 2π
r2
to find is the surface area of the cylindrical food storage container. Use 3. 14 for π. Round your answer to the nearest hundredth of a square inch
Answers
The surface area of the cylindrical food storage container given by the formula SA = 2πrh + 2πr² is 954. 56 inch².
The region is the space taken up by a flat, two-dimensional surface. Its unit of measurement is the square. A three-dimensional object's surface area is the area occupied by its outside surface. It is also measured in square units.
Surface area and volume are calculated for each geometric object in three dimensions. The surface area of an object refers to the space that it takes up.
As, we Know the Surface Area of Cylinder
= 2πr² + 2πrh
Radius of the base = 8 inches
Height of the cylinder = 11 inches.
Now, putting the values we get
= 2πr² + 2πrh
= 2 x 3.14 x 8 x 8 + 2 x 3.14 x 8 x 11
= 401.92 + 552.64
= 954. 56 inch²
Therefore, the surface area of the cylinder is 954. 56 inch².
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Complete question
Use the formula SA = 2πrh + 2πr2 to find is the surface area of the cylindrical food storage container. Use 3. 14 for π. Round your answer to the nearest hundredth of a square inch
A taxi drives at a speed of 40 kilometers (km) per hour. How far does it travel in 210 minutes?
Answers
The taxi travels 140 kilometers in 210 minutes at a speed of 40 km/h.
Let's calculate the distance a taxi travels in 210 minutes at a speed of 40 km/h.
Convert minutes to hours
Since the speed is given in km/h, we need to convert 210 minutes into hours.
There are 60 minutes in an hour, so divide 210 by 60:
210 minutes ÷ 60 = 3.5 hours
Calculate the distance
Now that we have the time in hours, we can use the formula for distance:
Distance = Speed × Time
In this case, the speed is 40 km/h, and the time is 3.5 hours.
Plug these values into the formula:
Distance = 40 km/h × 3.5 hours
Compute the result
Multiply the speed by the time to find the distance:
Distance = 140 km.
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Round 5 6/13 to the nearest whole number.
4
5
6
7
Answers
When approximating mixed fraction 5 6/13 to the nearest whole number, the rounded value is 5.
To round the mixed fraction 5 6/13 to the nearest whole number, we examine the fractional part, which is 6/13. The general rule for rounding mixed fractions is to consider the fractional part and round up if it is greater than or equal to 1/2, and round down if it is less than 1/2.
In this case, 6/13 is approximately 0.4615. Since it is less than 1/2, we need to round down to the nearest whole number. Therefore, when rounding 5 6/13 to the nearest whole number, the answer is 5.
A mixed fraction consists of a whole number part and a fractional part. When rounding a mixed fraction, we focus on the fractional part to determine the appropriate rounding direction. If the fractional part is exactly 1/2, it is typically rounded up to the next whole number.
However, in the case of 5 6/13, the fractional part is less than 1/2, so we round down. Rounding down gives us a more accurate approximation that is closer to the original value. In this instance, rounding 5 6/13 down to 5 provides a whole number estimate that is slightly smaller but still reasonably close to the initial mixed fraction.
Rounding serves as a useful tool in situations where precise values are not necessary and a simpler approximation is sufficient for practical purposes.
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What is the price per cubic inch for the regular size popcorn that’s base is - 5x3 inches height- 8 inches
and the volume is 187
Answers
The volume of a rectangular prism is given by the formula V = lwh, where l is the length, w is the width, and h is the height. In this case, we have:
V = 5 x 3 x 8
V = 120 cubic inches
The price of the popcorn is not given, so we cannot calculate the price per cubic inch.
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If a pair of jeans coast $14. 99 in 1973 when the CPI was 135, what would the price of jeans have been in 1995 if the CPI was 305
Answers
If the CPI was 305 in 1995, the price of jeans that cost $14.99 in 1973 would be approximately $47.05 in 1995 after adjusting for inflation.
To find the price of jeans in 1995, we first need to adjust the 1973 price for inflation using the Consumer Price Index (CPI). CPI measures the average change in prices of goods and services over time, so it can help us compare prices from different years.
First, we need to calculate the inflation rate between 1973 and 1995. We can do this by dividing the CPI in 1995 (305) by the CPI in 1973 (135):
Inflation rate = (305 / 135) * 100% = 226.67%
This means that prices in 1995 were about 2.27 times higher than in 1973. Now, we can apply this inflation rate to the price of jeans in 1973:
Price in 1995 = Price in 1973 * (1 + inflation rate)
Price in 1995 = $14.99 * (1 + 2.2667) = $47.05
Therefore, if the CPI was 305 in 1995, the price of jeans that cost $14.99 in 1973 would be approximately $47.05 in 1995 after adjusting for inflation. This calculation helps to compare the cost of goods across different time periods by taking inflation into account, thus giving a better understanding of the changes in purchasing power over time.
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Solve for x and y.
15)
4+18y
10x
10x-6
16y+6
N
L
M
Answers
The value of x and y is 11 and 4 respectively
What is cyclic quadrilateral?A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral.
A theorem in circle geometry states that the sum of opposite angles in a cyclic quadrilateral are supplementary. i.e they sum up to give 180.
10x + 16y+6 = 180
10x+16y = 174... eqn1
4+18y +10x-6 = 180
18y +10x = 182... eqn2
subtract equation 1 from 2
2y = 8
y = 8/2 = 4
Subtitle 4 for y in equation 1
10x+ 16(4)= 174
10x= 174-64
10x = 110
x= 110/10
x = 11
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Points E, F, G and H lie on the circle and EG = EH.
HF and EG intersect at K.
ET is a tangent to the circle at E.
Angle FET = 47° and angle FEG = 25°.
Find the value of x.
Answers
Using the inscribed angle theorem, the value of x in the circle shown is calculated as: m<x = 36°.
How to Apply the Inscribed Angle Theorem?According to the inscribed angle theorem an inscribed angle has an angle measure that is always half of the measure of the arc it intercepts.
Measure of arc EFG = 2(25 + 47) [inscribed angle theorem]
Measure of arc EFG = 144°
Measure of angle EHG = 1/2(m(EFG)) [inscribed angle theorem]
Substitute:
Measure of angle EHG = 1/2(144) = 72°
m<EHG = m<EGH = 72° [this is because triangle EHG is an isosceles triangle since sides EG = EH].
Therefore, we have:
m<x = 180 - 2(72)
m<x = 36°
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Complete the table to find the derivative of the function Original Function Rewrite 3 y = 2 (2x)-2 12 Differentiate Simplify 1 24x X
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The derivative of the function y = 2(2^x)-2 is 12 * 2^x ln(2) or 12ln(2)x(2^x-1).
To find the derivative of the function y = 2(2^x)-2, we will use the power rule and the chain rule of differentiation.
Apply the power rule to the function y = 2(2^x)-2. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).
y' = [2(2^x)-2]'
= 2[(2^x)-2]'
= 2ln(2^x)'
Apply the chain rule to (2^x)'. The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x))h'(x). In this case, g(x) = 2^x, so g'(x) = ln(2)*2^x.
y' = 2ln(2^x)'
= 2ln(2^x)
= 2ln(2)x(2^x-1)
Therefore, the derivative of the function y = 2(2^x)-2 is 12 * 2^x ln(2) or 12ln(2)x(2^x-1).
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Express the volume of the sphere x^2+ y^2 + z2 < 36 that lies between the cones z = √ 3x^2 + 3y^2 and z = √(x^2+y^2)/3
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The volume of the sphere that lies between the two cones is approximately 43.53 cubic units.
How to calculate the volume of the sphereTo find the volume of the sphere that lies between the given cones, we first need to determine the limits of integration.
Since the sphere has a radius of 6 (since x² + y² + z² = 36), we can use spherical coordinates to express the volume as an integral. Let's first consider the cone z = √3x² + 3y².
In spherical coordinates, this is equivalent to z = ρcos(φ)√3, where ρ is the radial distance and φ is the angle between the positive z-axis and the line connecting the origin to the point.
Similarly, the cone z = √(x²+y²)/3 can be expressed in spherical coordinates as z = ρcos(φ)/√3.
Since we're only interested in the volume of the sphere between these cones, we can integrate over the limits of ρ and φ that satisfy both inequalities.
The limits of ρ will be 0 (the origin) to 6 (the radius of the sphere).
To find the limits of φ, we need to solve for the intersection points of the two cones.
Setting the two equations equal to each other, we get:
ρcos(φ)√3 = ρcos(φ)/√3
Solving for φ, we get:
tan(φ) = 1/√3 Using the inverse tangent function, we find that: φ = π/6, 7π/6
So the limits of integration for φ will be π/6 to 7π/6.
Finally, we need to integrate over the full range of θ (the angle between the positive x-axis and the line connecting the origin to the point).
This will be 0 to 2π.
Putting it all together, the volume of the sphere between the two cones is:
∫∫∫ ρ^2sin(φ) dρ dφ dθ
With limits of integration:
0 ≤ ρ ≤ 6 π/6 ≤ φ ≤ 7π/6 0 ≤ θ ≤ 2π
Evaluating this integral gives:
V = 288π/5 - 216√3π/5
So the volume of the sphere that lies between the two cones is approximately 43.53 cubic units.
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What is the expected vale of an original investment of 3000 that has a 10% chance of ending up with a value of 2000
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The expected value of an original investment of 3000 that has a 10% chance of ending up with a value of 2000 is 2900. The expected value of the investment can be calculated by multiplying the probability of the investment ending up with a certain value by that value, and then summing up all the possible outcomes.
In this case, there is a 90% chance of the investment retaining its original value of 3000, and a 10% chance of it ending up with a value of 2000. To calculate the expected value, we can use the following formula:
Expected Value = (Probability of Outcome 1 × Value of Outcome 1) + (Probability of Outcome 2 × Value of Outcome 2)
Substituting the values,
Expected value = (0.9 x 3000) + (0.1 x 2000)
Expected value = 2700 + 200
Expected value = 2900
Therefore, the expected value of the original investment of 3000 is 2900.
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