In the diagram below of circle O, PA is tangent to circle O
at A, and PBC is a secant with points B and C on the circle.
A
4
8
B
C
If PA= 8 and PB = 4, what is the length of BC?
The length of BC is 6 units.
Describe Tangent?In geometry, the tangent is commonly used to describe the relationship between a circle and a straight line. A tangent line to a circle is a line that intersects the circle at exactly one point. This point of intersection is called the point of tangency. The tangent line is perpendicular to the radius of the circle at the point of tangency.
By the tangent-secant theorem, we have:
PB^2 = PA(PA + AB)
where AB is the length of the tangent segment from point A to the point of intersection of the secant with the circle.
Substituting the given values, we get:
4^2 = 8(8 + AB)
Simplifying and solving for AB, we get:
16 = 64 + 8AB
8AB = -48
AB = -6
Since AB is a length, it must be positive, so we take the absolute value:
AB = 6
Now we can use the secant-tangent theorem to find the length of BC:
PC(PC + BC) = PB^2
Substituting the given values and using the fact that PC = PA = 8, we get:
8(8 + BC) = 4^2
64 + 8BC = 16
8BC = -48
BC = -6
Again, since BC is a length, it must be positive, so we take the absolute value:
BC = 6
Therefore, the length of BC is 6 units.
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To make cherry trail mix, Sebastián needs 4 oz of nuts for every 3 oz of dried cherries. To make sunflower trail mix, he needs 5 oz of nuts for every 2 oz of sunflower seeds. Can Sebastian make more cherry trail mix or sunflower trail mix with 20 oz of nuts? Show your work.
Answer:
Cherry Trail Mix
Step-by-step explanation:
cherry trail mix ratio = 4:3
sunflower trail mix ration = 5:2
4 goes into 20 5 times so multiply 3 by 5 to get 15. Add them together to make 35 oz
5 goes into 20 4 times so multiply w by 4 to get 8. Add them together to make 28 oz
35 > 28
Penny says that 4 x 65 = 260. Estimate to check pennys answer. Is she right? Explain
Edgar has $7,000 in an account that earns 15% interest compounded annually. To the nearest cent, how much interest will he earn in 3 years?
Answer:
429
Step-by-step explanation:
Answer:
Edgar will earn $3,646.30 in interest over 3 years.
Step-by-step explanation:
I'd be happy to walk you through the problem step by step!
The problem is asking how much interest Edgar will earn in 3 years on an initial deposit of $7,000 that earns 15% interest compounded annually.
To solve this problem, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount including interest
P = the principal amount (the initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the time (in years) the money is invested
We are given that:
P = $7,000
r = 15% = 0.15 (as a decimal)
n = 1 (compounded annually)
t = 3 years
Now, we can plug these values into the formula and solve for A, which is the final amount including interest:
A = $7,000(1 + 0.15/1)^(1*3)
We simplify the expression inside the parentheses first, by dividing the annual interest rate by the number of times the interest is compounded per year:
A = $7,000(1.15)^(3)
We raise 1.15 to the power of 3 using a calculator or by multiplying 1.15 by itself 3 times:
A = $7,000(1.5209)
We multiply the initial deposit by the final amount including interest to get the total amount of interest earned:
Interest = $7,000(1.5209) - $7,000
We simplify the expression:
Interest = $10,646.30 - $7,000
Interest = $3,646.30
So, Edgar will earn $3,646.30 in interest over 3 years.
Hope this helped! If it didn't, I'm sorry! If you still need more help on this, ask me! :]
The explicit formula of an arithmetic sequence is an=−6−4(n−1). What is the recursive formula of the sequence? ( 1 point)
A. an+1=an−4, a1=−6
B.an+1=an−6, a1=−4
C. an+1=an+4, a1=−6
D. an+1=an−4
Answer: Option A. an+1=an-4, a1=-6
Step-by-step explanation:
Which pair of angles are corresponding?
A. <1 and <3
B. <3 and <4
C. <1 and <8
D. <2 and <6
Answer:
2 and 6
Step-by-step explanation:
2 and 6 is the correct option
Would the interval found in part c increase, decrease, or remain the same if fewer than 95 people were surveyed? justify your answer
The confidence interval would increase because with fewer people, the margin of error would be larger.
The confidence interval found in part c would increase if fewer than 95 people were surveyed. This is because the margin of error is calculated using the sample size; the smaller the sample size, the larger the margin of error. The margin of error dictates the range of values that the population parameter is likely to fall within. Therefore, if fewer people are surveyed, the margin of error would increase, resulting in a wider confidence interval. This means that the interval from part c would no longer be valid if the sample size decreased, and a new interval with a larger range would need to be calculated.
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Solve the equation.
8.88=4.44(x−7)
x = ?
Answer: x=9
Step-by-step explanation:
Simplify the expression
Add 777 to both sides
Simplify the expression
Divide both sides by the same factor
Simplify the expression
Neil and Joey share a 24-ounce box of cereal. By the end of the week, Neil has eaten 3 8 of the box, and Joey has eaten 1 4 of the box of cereal. How many ounces are left in the box?
Neil and Joey ate a total of 15 ounces of cereal from the 24-ounce box. By subtracting the amount eaten from the total amount, we find that there are 9 ounces of cereal left in the box.
Neil has eaten 3/8 of the box, which is equal to (3/8) x 24 = 9 ounces.
Joey has eaten 1/4 of the box, which is equal to (1/4) x 24 = 6 ounces.
Together, they have eaten 9 + 6 = 15 ounces.
Subtracting the amount they have eaten from the total amount, we get:
24 - 15 = 9
Therefore, there are 9 ounces of cereal left in the box.
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Give a linear equation for the total remaining oil reserves, , in terms of , the number of years since now. (Be sure to use the correct variable and Preview before you submit. ) R=
A. The linear equation is R = -18t + 1,820
B. total reserve oil 1,568 billions of barrels
C. approximately 101.11 years
To make our equation, we'll use the form R = mt + b. M represents how many billion barrels of oil are being lost each year, which we know is 18 billion. So -18 will be our m. B is how many total barrels of oil there are, which is 1,820. So 1,820 will be our b. Now the equation looks like this:
R = -18t + 1,820
We can use this equation to answer Part B.
Replace the t with 14:
R = -18(14) + 1,820
Now solve for R:
R = -18(14) + 1,820
R = -252 + 1,820
R = 1,568
14 years from now, there will be 1,568 billions of barrels left.
To solve part C, we need to find how many years it will take for all of the oil to be used up. After it's all used up, the total amount of oil will be 0, so we can replace R with 0 and then solve for t:
0 = -18t + 1,820
Subtract 1,820 from both sides to isolate -18t:
0 - 1,820 = -18t + 1,820 - 1,820
-1,820 = -18t
Divide both sides by -18 to isolate the t:
-1,820/-18 = -18t/-18
101.11 = t
After approximately 101.11 years, all of the oil will be used up.
The complete question is-
Suppose that the world's current oil reserves is R = 1820 billion barrels. If, on average, the total reserves
is decreasing by 18 billion barrels of oil each year, answer the following:
A.) Give a linear equation for the total remaining oil reserves, R, in billions of barrels, in terms of t, the
number of years since now. (Be sure to use the correct variable and Preview before you submit.)
R
B.) 14 years from now, the total oil reserves will be
billions of barrels.
C.) If no other oil is deposited into the reserves, the world's oil reserves will be completely depleted (all
used up) approximately
years from now.
(Round your answer to two decimal places.)
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Which of the integers are multiples of 40 ? Select all that apply. A. 10,
B. 160 C. 360
D. 20 E. 580
Answer:
B AND C
Step-by-step explanation:
B AND C ARE THE CORRECT MULTIPLES 40
The area of a triangle is one fourth the area of a square. If the base of the triangle and a
side of the square are equal, what is the ratio of the side of the square to the height of the
triangle?
Answer:
see below
Step-by-step explanation:
2:1
Look at the image below.
Please help ASAP!
Filipe bought a classic car, originally valued at $60,000, in 2012. He believes that the value of the car depreciates exponentially at a rate of 4% each year.
Round the expected value of Filipe's car after 6 years to the nearest cent.
Click “show your work” and show your steps to solve this problem.
(Hint: You need to use the exponential decay formula)
In a case whereby Filipe bought a classic car, originally valued at $60,000, in 2012. whee the value of the car depreciates exponentially at a rate of 4% the expected value of Filipe's car after 6 years is $46970
How can the expected value of Filipe's carbe calculated?Based on the given information, the future value when there is a depreciation can be calculated using the formular below:
FV =[ P (1 - r)^n]
where the
R represent the rate of depreciation =4% = 0.04
N represent the number of years = 6 years
FV represent the Future value
P represent the Present value
From the formula we can input the values as :
$60,000 x ( 1 - 0.04)^6 = $42,890
=$46965.5
=$46970 (rounded up)
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The length of a rectangle is twice as long as it’s width. it’s perimeter is 18 in. How long and how wide is the rectangle?
Answer: Let's start by using algebra to solve the problem.
Let's call the width of the rectangle "w". Then we know that the length of the rectangle is twice the width, or "2w".
The formula for the perimeter of a rectangle is:
P = 2l + 2w
where P is the perimeter, l is the length, and w is the width.
We know that the perimeter is 18 inches, so we can substitute the values we have:
18 = 2(2w) + 2w
Simplifying:
18 = 4w + 2w
18 = 6w
Dividing both sides by 6:
w = 3
So the width of the rectangle is 3 inches.
To find the length, we can use the fact that it is twice the width:
l = 2w = 2(3) = 6
So the length of the rectangle is 6 inches.
Therefore, the rectangle is 6 inches long and 3 inches wide.
enjoy!
The ordered pair (a,b) satisfies the inequality y
Answer:
If you add 5 to a, it will be greater than b
Step-by-step explanation:
i got it right
While trying to understand and prepare data for further analysis you find out that there are small amount missing values for a categorical variable, and you know that keeping this variable is important for the success of your project. What would be the correct way to handle the missing values?
A. Replace missing values with the mean value.
B. Although it seems like keeping this variable is important for the success of the project, due to missing values decide to exclude this variable from the analysis.
C. Replace the missing values with the most frequent category.
D. Do nothing and proceed with further analysis.
The result, doing nothing is not a sensible approach to missing data handling.
The correct way to handle the missing values is to replace the missing values with the most frequent category.What is a variable?A variable is any feature, number, or measurement that may be measured, manipulated, or controlled in an experiment. Variables in scientific investigations are classified as dependent or independent. A categorical variable is one that may take on one of several different categories or distinct groups of data. Understanding the problem statement While attempting to understand and prepare data for further analysis, you discover that there are a small number of missing values for a categorical variable.
Since you know that retaining this variable is important for the project's success, what is the appropriate method for handling the missing values?Missing values should be replaced with the most frequent category because this is the best way to handle them.
To get a complete dataset, the missing values must be filled. The variables with missing data are imputed to improve the accuracy of your analysis.Replace missing values with the mean value (option A) - This technique is only applicable to quantitative variables.
In categorical variables, it makes no sense to replace missing data with the mean value. It might produce nonsensical data, making the whole data set unusable.Although it seems like keeping this variable is important for the success of the project, due to missing values decide to exclude this variable from the analysis (option B) - This alternative may lead to the exclusion of crucial data that can help with a future examination.
Replace the missing values with the most frequent category (option C) - The most frequent category is frequently used to replace missing data. The most common category is determined and substituted for missing data. It's a reasonable method because the data is still valuable and the categorical variable is critical for the project's success.Do nothing and proceed with further analysis (option D) - The examination results would be influenced if we use the existing data without filling in the missing information. As a result, doing nothing is not a sensible approach to missing data handling.
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PLEASE HELP!! ⚠️⚠️⚠️
Bernie wanted to stock up on drinks for an upcoming party. first, he spent $37 on 7 cases of juice and 8 cases of soda, which is all the store had in stock. A few days later, he returned to the store and purchased an additional 7 cases of juice and 15 cases of soda, spending a total of $51. What is the price of each drink?
The price of each drink is given as follows:
Case of juice: $3.Case of soda: $2.How to obtain the price of each drink?The price of each drink is obtained by a system of equations, for which the variables are given as follows:
Variable x: cost of a case of juice.Variable y: cost of a case of soda.He spent $37 on 7 cases of juice and 8 cases of soda, hence:
7x + 8y = 37.
Purchased an additional 7 cases of juice and 15 cases of soda, spending a total of $51, hence:
7x + 15y = 51.
Hence the system is composed by these two equations:
7x + 8y = 37.7x + 15y = 51.Subtracting the second equation by the first, the value of y is obtained as follows:
7y = 14
y = 2.
Then the value of x is obtained as follows:
7x + 15(2) = 51
7x = 21
x = 21/7
x = 3.
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Apple stock rose from $75 per share on January 1, 2013 to $109 per share on January 1, 2015. During the
same period, Microsoft stock rose from $27 to $47.
(a) What was the absolute change in dollars in the Apple stock price?
(b) What was the absolute change in dollars in the Microsoft stock price?
(c) What was the relative change in the Apple stock price, rounded to the nearest percent? $
(d) What was the relative change in the Microsoft stock price, rounded to the nearest percent?
%
(e) Suppose you invested $5,000 in Apple stock on January 1, 2013. How many whole shares would you have
been able tobuy? (Enter a whole number.)
shares
%
(f) At the end of the two years, if you sold all these shares, what would be the total amount in dollars from
the sale? $
(g) Suppose you invested $5,000 in Microsoft stock on January 1, 2013. How many whole shares would you
have been able to buy? (Enter a whole number.)
shares
(h) At the end of the two years, if you sold all these shares, what would be the total amount from the
sale? $
The stock price of corporation is [tex]45[/tex]%, [tex]34[/tex] and [tex]20[/tex] correspondingly for the absolute change in the dollar and the Ms dollar.
The price of a stock right now.It is the latest trading price for a stock or some other security. Inside an market, the place to start is the current price. It demonstrates the cost that a both parties would be willing to pay for a subsequent deal involving that security.
(a) The absolute change in dollars in the Apple stock price is:
[tex]109 - 75 = 34[/tex]
(b) The absolute change in dollars in the Microsoft stock price is:
[tex]47 - 27 = 20[/tex]
(c) The relative change in the Apple stock price, rounded to the nearest percent is: [tex]((109 - 75)/75) * 100 = 45[/tex]%
(d) The relative change in the Microsoft stock price, rounded to the nearest percent is: [tex]((47 - 27)/27) * 100 = 74[/tex]%
(e) If you invested $[tex]5,000[/tex] in Apple stock on January 1, 2013, and the price per share was $[tex]75[/tex], you could buy: [tex]5,000/75 = 66.67[/tex] shares
Rounded down to the nearest whole number, you would be able to buy [tex]66[/tex] shares.
(f) If you sold all [tex]66[/tex] shares of Apple stock at the price of $[tex]109[/tex] per share at the end of two years, the total amount from the sale would be:
[tex]66 shares * $109/share = 7,194[/tex]
(g) If you invested $[tex]5,000[/tex] in Microsoft stock on January 1, 2013, and the price per share was $[tex]27[/tex], you could buy: [tex]5,000/27 = 185.19[/tex] shares
Rounded down to the nearest whole number, you would be able to buy [tex]185[/tex] shares.
(h) If you sold all [tex]185[/tex] shares of Microsoft stock at the price of $[tex]47[/tex] per share at the end of two years, the total amount from the sale would be:
[tex]185 shares * 47/share = 8,695[/tex]
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write an equation of the line that passes through (-1,0) and (0,0)
y=
Answer:
y = 0
Step-by-step explanation:
the 2 points (- 1, 0 ) and (0, 0 ) both have the same y- coordinate.
this indicates the line is horizontal with equation
y = c ( c is the value of the y- coordinates the line passes through )
both points have a y- coordinate of 0 , then
y = 0 ← equation of line
NO EXPLANATION JUST ANSWER!
Answer:
405mm^3
Hope this helps!
Brainliest and a like is much appreciated!
Answer:
405
Step-by-step explanation:
Answer is above!1. Evaluate each expression. Give your answer in radians:
i. Without using a calculator: cos^-1(-√2/2)=
ii. Use a calculator: sin^-1(-0.512)=
i. π/4
ii. -0.529
1. Evaluate each expression. Give your answer in radians:
i. Without using a calculator: cos⁻¹(−√2/2)Firstly, let us discuss the meaning of the inverse cosine function. The inverse cosine function is the angle (in radians) that corresponds to the cosine value of the input angle. The inverse cosine function returns values in the range [0, pi].cos⁻¹(−√2/2) can be read as "what angle has a cosine of -√2/2?"In a right-angled triangle with angle θ, adjacent is -1, hypotenuse is √2, and opposite is -1. Therefore, we can write the cosine ratio as adjacent over hypotenuse or -1/√2. Let us now solve for θ using the formula for cosine:cosθ = adjacent/hypotenusecosθ = -1/√2θ = cos⁻¹(-1/√2)The exact value of cos⁻¹(-1/√2) is π/4. Therefore, cos⁻¹(-√2/2) = π/4.
ii. Use a calculator: sin^-1(-0.512)As previously mentioned, the inverse sine function is the angle (in radians) that corresponds to the sine value of the input angle. The inverse sine function returns values in the range [-π/2, π/2].sin⁻¹(-0.512) can be read as "what angle has a sine of -0.512?" To find this value, we use a calculator. First, we press the "sin⁻¹" button or "2nd" + "sin" button. Then, we enter the value we want to find, which is -0.512. The final answer is -0.529. Therefore, sin⁻¹(-0.512) = -0.529.
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Solve for x. Round your final answer to the nearest tenth.
Law of cosines.
55 90 50
Answer: the length of side c is approximately 74.3 units.
Step-by-step explanation: In a triangle with sides a, b, and c, and opposite angles A, B, and C, respectively, the Law of Cosines states that:
c^2 = a^2 + b^2 - 2ab*cos(C)
We are given the following information:
a = 55 (the side opposite angle A)
b = 50 (the side opposite angle B)
C = 90 degrees (the angle opposite side c)
Substituting these values into the Law of Cosines, we get:
c^2 = 55^2 + 50^2 - 2(55)(50)*cos(90)
c^2 = 3025 + 2500 - 0
c^2 = 5525
Taking the square root of both sides, we get:
c = sqrt(5525)
c ≈ 74.3
PLEASE SOMEBODY HELP
Step-by-step explanation:
A square has all the sides equal
If AD which is a side is 11, all other sides are 1
BC is 11
AC is a diagonal that divides the square and forms a triangle
AD =11
CD =11
Use Pythagoras theorem to find AC which is hypotenuse
AC² = AD² + CD²
= 121 + 121
=242
If AC² = 242, then AC is square root of 242
AC = 15.56 (approximate if required) 15.6 to 1d.p if required
BD = AC = 15.56
EC is half of BD and AC = 15.56/2
EC = 7.78 (check the decimal place required) 7.8 to 1d.p if required
Angle DAB is 90
Angle AEB is 90
Angle CBD is 45
Angle BAC is 45
(Add the units)
Quadrilateral C with 4 congruent sides is a square, all opposite sides are congruent
Quad L is a parallelogram, consecutive angles are supplementary and opposite angles are congruent
Kusum brought 50 glasses for Rs 1500 out of which 4 were broken. If she sold
the rest of the glasses for Rs 35 each, how much profit or loss precent did she bear?
Answer: 7.33%
Step by step explanation:
CP of 50 glasses = Rs. 1500
4 glasses broken
Glass remaining = 50 - 4 = 46
Selling price of 1 glass = Rs 35
SP of 46 glass = 46 x 35 = Rs 1610
Profit = SP - CP = 1610 - 1500 = 110
Profit percentage,
= Profit/CP x 100%
= 110/1500 x 100%
= 7.33%
The cost price of 50 glasses = Rs.1500
The remaining glasses = 46
The Selling price of remaining glasses = 46 × 35
=Rs. 1610
Now
Profit = 1610 - 1500
= Rs.110
Now
Profit Percent
[tex] \frac{Profit}{CP} \times 100%[/tex]
[tex] \frac{110}{1500} \times 100%[/tex]
= 7.33%
What is the range of f(x)? {x | −2 ≤ x < 4} {x | −2 < x ≤ 4} {y | −5 < y < −1} {y | −5 ≤ y ≤ −1}
Option (d) is correct i.e., y | −5 ≤ y ≤ −1 , for this we have to know the concept of range and coordinates.
What is Range?In mathematics, a function's range can refer to one of two closely related ideas: The function's codomain a representation of the action A binary relation f between two sets P and Q is a function if there is exactly one y in Q such that f connects x to y for every x in P .
In geometry, a coordinate system is a method for determining how to place points or other geometrical objects on a manifold, such Euclidean space, uniquely using one or more numbers, or coordinates.
The range of a function is the set of all values that f can produce for all the x-axis in the domain.
If we are given the graph, in order to find the range, we project the graph into the y-axis. Informally, we draw the "shadow" of the graph into the y axis as in the FIGURE attached.
Option (d) is correct i.e., y | −5 ≤ y ≤ −1 .
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Find all complex zeros of the given polynomial function, and write the polynomial in completely factored form.
f(x)=4x^3+3x^2-44x-33
Find the complex zeros of f. Repeat any zeros if their multiplicity is greater than 1.
x=
From the given information, the complex zeros of the polynomial are x = -3/4, (9 + 17i)/8, (9 - 17i)/8.
To find the complex zeros of the given polynomial function, we can use the Rational Root Theorem to check for possible rational roots, and then use synthetic division or long division to factor the polynomial.
The possible rational roots of the polynomial are of the form ±a/b, where a is a factor of the constant term 33 and b is a factor of the leading coefficient 4. Therefore, the possible rational roots are:
±1/4, ±3/4, ±1/2, ±3/2, ±11/4, ±33/4, ±11/2, ±33/2.
We can check these possible roots using synthetic division, and we find that the polynomial has a rational root x = -3/4. Dividing by (x + 3/4) using synthetic division, we get:
4x³ + 3x² - 44x - 33 = (x + 3/4)(4x² - 9x - 44)
Now, we can use the quadratic formula to find the roots of the quadratic factor 4x² - 9x - 44:
x = (9 ± √(9² + 4(4)(44)))/(2(4)) = (9 ± 17i)/8.
To factor the polynomial completely, we can use the complex zeros and the linear factor we found earlier:
f(x) = 4x³ + 3x² - 44x - 33 = (x + 3/4)(x - (9 + 17i)/8)(x - (9 - 17i)/8).
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Games A group of friends tries to keep a beanbag from touching the ground.
On one kick, the beanbag's height can be modeled by h = -161² + 14t + 2, where h
is the height in feet above the ground and t is the time in seconds. Find the time it
takes the beanbag to reach the ground.
The time it takes for the beanbag to reach the ground is t = 1 second.
What is quadratic equation ?
A quadratic equation is a expression of the form ax^{2} + bx + c = 0, where x is the variable, and a, b, and c are constants. It is a type of second-degree polynomial equation, which can be solved using various methods, including factoring, completing the square, and using the quadratic formula
We know that the beanbag will touch the ground when its height, h, equals zero. Therefore, we can set the equation for h equal to zero and solve for t:
h = -16t² + 14t + 2
0 = -16t² + 14t + 2 (substitute h=0)
0 = -8t² + 7t + 1 (divide both sides by 2)
0 = (t-1)(-8t-1) (factor the quadratic)
So, either t - 1 = 0 or -8t - 1 = 0. Solving for t in each case, we get:
t - 1 = 0 => t = 1
-8t - 1 = 0 => t = -1/8
The negative value of t doesn't make sense in this context, since time cannot be negative. Therefore, the time it takes for the beanbag to reach the ground is t = 1 second.
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Here's what she knows:
The distance between the center of the cow pen and the center of the horse pen is 15 yards.
The distance between the center of the cow pen and the center of the pig pen is 10 yards.
All radii are whole numbers, and all are greater than 3 yards.
● The distance between the center of the cow pen and the center of the horse pen is 15 yards
No two pens are the same size.
She has labeled distances between the barn & points of tangency to the pens in her diagram.
You can help her decide which animal goes in which pen.
1. Now help her find the distance from the barn to the center of each pen! Show all of your work.
Use the page below for workspace as needed.
Distance from barn to center of...
Horse pen:
Cow pen:
Pig pen:
9.5 yds.
8.75 yds.
OX
112⁰
248°
8.75 yds.
16 yds.
This means that the distance from the barn to the center of the pig pen is 8.75 yards.
What is distance?Distance is the amount of space between two points or objects. It is commonly measured in linear units such as meters, feet, or miles. Distance is a fundamental concept in mathematics, physics, and other sciences. It is used to measure the size of objects, the speed of objects, and the time it takes for objects to move. Distance can also be used to measure the strength of a force, the intensity of a light source, or the similarities between two objects. Distance is a crucial concept for understanding the physical world, as it helps us to understand the relationships between objects and how they interact.
The difference in the distances between the center of the cow pen and the center of the horse pen compared to the center of the cow pen and the center of the pig pen is due to the different radii of the pens. Since all radii are greater than 3 yards, the distance of the cow pen and the horse pen is longer than the distance of the cow pen and the pig pen.
To calculate the distance from the barn to the center of each pen, we need to use the angle of tangency and the radii of the pens. For each pen, we need to calculate the angle of tangency, which is the angle between the hypotenuse of the triangle formed by the barn, the pen and the center of the pen and the radius of the pen. This can be calculated by either the Law of Cosines or the Law of Sines.
For the horse pen, the angle of tangency is 112 degrees and the radius is 7 yards. This means that the distance from the barn to the center of the horse pen is 9.5 yards. For the cow pen, the angle of tangency is 248 degrees and the radius is 5 yards. This means that the distance from the barn to the center of the cow pen is 8.75 yards. Finally, for the pig pen, the angle of tangency is 128 degrees and the radius is 3 yards. This means that the distance from the barn to the center of the pig pen is 8.75 yards.
Therefore, the difference in the distances between the center of the cow pen and the center of the horse pen compared to the center of the cow pen and the center of the pig pen is due to the different radii of the pens. The larger the radius of the pen, the longer the distance from the barn to the center of the pen.
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The distances from the barn to the center of the Horse pen, Cow pen, and Pig pen are 15 yards, 15 yards, and 18.35 yards respectively.
What is distance?Distance is the amount of space between two points or objects. It is commonly measured in linear units such as meters, feet, or miles. Distance is a fundamental concept in mathematics, physics, and other sciences. It is used to measure the size of objects, the speed of objects, and the time it takes for objects to move.
First, let's start with the Horse pen. To find the distance from the barn to the center of the Horse pen, we'll use the law of cosines. The law of cosines states that for a triangle ABC, with angle A being 90°, the following equation can be used to find the length of side a:
a² = b²+ c² - 2bc cosA
Therefore, for our triangle, with angle A being 90°, we have:
a²= 15² + 10² - 2(15)(10)cos90
a²= 225 + 100 - 300
a²= 25
a = √25
a = 5
Now, we can find the distance from the barn to the center of the Horse pen by adding the distance from the center of the pen to the barn, which is 10 yards, to 5 yards, which is the distance from the barn to the center of the Horse pen.
Therefore, the distance from the barn to the center of the Horse pen is 10 + 5 = 15 yards.
Now, let's find the distance from the barn to the center of the Cow pen. Again, we'll use the law of cosines. The law of cosines states that for a triangle ABC, with angle A being 90°, the following equation can be used to find the length of side a:
a² = b²+ c² - 2bc cosA
Therefore, for our triangle, with angle A being 90°, we have:
a²= 15² + 9.5² - 2(15)(9.5)cos90
a²= 225 + 90.25 - 285.75
a²= 30.25
a = √30.25
a = 5.5
Now, we can find the distance from the barn to the center of the Cow pen by adding the distance from the center of the pen to the barn, which is 9.5 yards, to 5.5 yards, which is the distance from the barn to the center of the Cow pen.
a²= b²+ c²- 2bc cosA
Therefore, for our triangle, with angle A being 90°, we have:
a²= 16² + 8.75² - 2(16)(8.75)cos90
a²= 256 + 76.5625 - 240
a² = 92.5625
a = √92.5625
a = 9.6
Now, we can find the distance from the barn to the center of the Pig pen by adding the distance from the center of the pen to the barn, which is 8.75 yards, to 9.6 yards, which is the distance from the barn to the center of the Pig pen.
Therefore, the distance from the barn to the center of the Pig pen is 8.75 + 9.6 = 18.35 yards.
In conclusion, the distances from the barn to the center of the Horse pen, Cow pen, and Pig pen are 15 yards, 15 yards, and 18.35 yards respectively.
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The radius of a quarter circle is 1 yard. What is the quarter circle's perimeter?
The perimeter of the quarter circle with radius 1 yard is approximately 2.57 yards.
To find the perimeter of a quarter circle, we need to add up the lengths of its curved edge and its straight edge.
In this case, the curved edge is a quarter of a circle with radius 1 yard. The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle. Since we only need a quarter of the circumference, we can divide the formula by 4:
C = (2πr)/4
Substituting r = 1, we get:
C = (2π(1))/4
C = π/2
Therefore, the curved edge of the quarter circle has a length of π/2 yards. The straight edge of the quarter circle is simply the radius, which is also 1 yard.
So the perimeter of the quarter circle is the sum of the curved edge and the straight edge:
Perimeter = (π/2) + 1
Using 3.14 for π and adding the two terms, we get:
Perimeter ≈ 2.57 yards
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A slot machine takes only one-dollar and two-dollars coins and contains a total of thirty six coins altogether. The value of the coins in the machine is ????$55. 4.3.1 Model the statement into two linear equations by letting the number of one-dollar coins to be x and the number of two-dollar coins to be y . (4) 4.3.2 Solve the two equations you formed in 4.3.1 using Cramer’s rule to find the number of coins of each value. (6)
There are 17 one-dollar coins and 19 two-dollar coins in the machine.
How did we arrive at these values?4.3.1 Model the statement into two linear equations by letting the number of one-dollar coins to be x and the number of two-dollar coins to be y:
Let x be the number of one-dollar coins and y be the number of two-dollar coins in the machine. Then we can form the following two linear equations:
Equation 1: x + y = 36 (total number of coins)
Equation 2: 1x + 2y = 55 (total value of coins)
4.3.2 Solve the two equations you formed in 4.3.1 using Cramer’s rule to find the number of coins of each value:
To solve the two equations using Cramer's rule, we need to write them in matrix form:
| 1 1 | | x | | 36 |
| | x | | |
| 1 2 | | y | | 55 |
Using Cramer's rule, we first calculate the determinant of the coefficient matrix:
D = | 1 1 |
| 1 2 |
D = (1 x 2) - (1 x 1) = 1
Then we replace the first column of the coefficient matrix with the constants from the right-hand side of the equations:
Dx = | 36 1 |
| 55 2 |
And calculate the determinant of this matrix:
Dx = (36 x 2) - (55 x 1) = 17
Similarly, we replace the second column of the coefficient matrix with the constants from the right-hand side of the equations:
Dy = | 1 36 |
| 1 55 |
And calculate the determinant of this matrix:
Dy = (1 x 55) - (1 x 36) = 19
Finally, we can calculate the values of x and y using the formulas:
x = Dx / D = 17 / 1 = 17
y = Dy / D = 19 / 1 = 19
Therefore, there are 17 one-dollar coins and 19 two-dollar coins in the machine.
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