Answer:
The area of a circle is calculated using the formula A = πr^2, where r is the radius of the circle. Since the diameter is given, we can divide it by 2 to get the radius.
Given:
Diameter = 16 ft
Radius = Diameter / 2 = 16 / 2 = 8 ft
Now, we can plug in the value of the radius into the formula for area:
A = πr^2
A = π(8^2)
A = π(64)
A ≈ 200.96 (rounded to two decimal places)
Therefore, the area of the circle to the nearest whole number is 201 square feet.
Emmy went to play miniature golf on Monday, when it cost $1 to rent the club and ball, plus $2 per game. Liam went Thursday, paying $1 per game, plus rental fees of $5. By coincidence, they played the same number of games for the same total cost. How many games did each one play?
Emmy and Liam each played 4 games according to the given statement.
What is an equation?An equation is a claim that two expressions are equal, typically indicated by the equals symbol (=). In mathematics, equations are used to simulate real-world scenarios, solve problems, and depict relationships between variables.
Exponents, logarithms, and trigonometric functions can all be used in equations, in addition to basic operations like addition, subtraction, multiplication, and division.
Let us suppose the number of games played = x.
Thus, for Emmy we have:
E = 1 + 2x
For Liam the equation is:
L = 5 + 1x
Equating the two equations we have:
1 + 2x = 5 + 1x
x = 4
Hence, Emmy and Liam each played 4 games according to the given statement.
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During a flood, there were 6000 acres of land under water. After 2 days, only 3375 acres of land were under water. Assume that the water receded at an exponential rate. Write a function to model this situation that has a B-value of 1.
where t is measured in days, and A(t) represents the amount of flooded land at time t. This function has a B-value of -0.3118.
To model the situation of the flood, we can use an exponential decay function, which represents the decreasing amount of flooded land over time. The function can be written as:
[tex]A(t) = A0 * e^{(-kt)}[/tex]
where A(t) is the amount of flooded land at time t, A0 is the initial amount of flooded land, k is a constant representing the rate of decay, and e is the mathematical constant approximately equal to 2.718.
To determine the value of k, we can use the given information that after 2 days, only 3375 acres of land were under water. Substituting t = 2 and A(t) = 3375 into the equation above, we get:
[tex]3375 = A0 * e^{(-2k)[/tex]
We also know that initially, there were 6000 acres of land under water. Substituting A0 = 6000 into the equation above, we get:
Dividing both sides by 6000, we get:
ln(0.5625) = -2k[tex]ln(0.5625) = -2k[/tex]
Taking the natural logarithm of both sides, we get:
[tex]ln(0.5625) = -2k[/tex]
Solving for k, we get:
[tex]k = -ln(0.5625)/2[/tex]
k ≈ 0.3118
Therefore, the function to model the situation of the flood is:
[tex]A(t) = 6000 * e^{(-0.3118t)}[/tex]
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STRUCTURE The ratio of circumference to diameter is the same for every circle. Is the ratio of circumference to radius the same for every circle? Make sure to explain!
No, the ratio of circumference to radius is not the same for every circle.
What is ratio?Ratio refers to the quantitative relation between two or more values, typically expressed in the form of a fraction or a proportion.
According to given information:No, the ratio of circumference to radius is not the same for every circle. The ratio of circumference to diameter, also known as pi (π), is a constant value that remains the same for every circle. It is approximately equal to 3.14 or 22/7. However, the ratio of circumference to radius varies depending on the size of the circle.
The formula for circumference of a circle is C=2πr, where C is the circumference and r is the radius. Therefore, the ratio of circumference to radius is C/r = 2π. This means that for circles of different sizes, the ratio of circumference to radius will differ since the value of pi remains the same while the radius changes.
For example, if we consider two circles, one with a radius of 2 cm and the other with a radius of 4 cm, the ratio of circumference to radius for the first circle will be 2π (since C = 2πr = 2π x 2 = 4π) and for the second circle, it will be 2π (since C = 2πr = 2π x 4 = 8π). Thus, the ratio of circumference to radius is not the same for every circle.
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true or false: a linear programming problem can have an optimal solution that is not a corner point. select one: true false
It is true that a linear programming problem can have an optimal solution that is not a corner point.
How given statement is true? Explain further?In linear programming, the optimal solution represents the point where the objective function is optimized while still satisfying all the constraints.
In some cases, the optimal solution may occur at a corner point of the feasible region, where two or more of the constraints intersect.
However, it is possible for the optimal solution to occur at a point that is not a corner point, but rather lies on an edge or a line segment of the feasible region.
This can occur when the objective function is parallel to one of the constraint lines or when there are redundant constraints that limit the feasible region.
Therefore, it is true that a linear programming problem can have an optimal solution that is not a corner point.
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Find the points on the surface z2 = xy +16 closest to the origin. The points on the surface closest to the origin are (Type an ordered triple. Use a comma to separate answers as needed. )
The points on the surface z² = xy + 16 closest to the origin are: (-4,4,0) and (4, -4, 0)
We know that the distance between an arbitrary point on the surface and the origin is d(x, y, z) = √(x² + y² + z²)
Using Lagrange multipliers,
L(x, y, z, λ) = x² + y² + z² + λ(z² - xy - 16)
We have partial derivatives.
[tex]L_x[/tex] = 2x - λy
[tex]L_y[/tex] = 2y - λx
[tex]L_z[/tex] = 2z + 2zλ
[tex]L_\lambda[/tex] = z² - xy - 16
Now we set each partial derivative to zero to find critical points.
[tex]L_x[/tex] = 0
2x - λy = 0
[tex]L_y[/tex] = 0
2y - λx = 0
After solving above equations simultaneously we get (x + y)(x - y) = 0
i.e., x = -y OR x = y
[tex]L_z[/tex] = 0
2z + 2zλ = 0
z = 0 OR λ = 0
Consider [tex]L_\lambda[/tex] = 0
z² - xy - 16 = 0
-xy = 16 ............(as z = 0)
when x = y then -y² = 16 which is not true.
So, consider x = -y
-(-y)y = 16
y² = 16
y = ±4
when y = 4 then we get x = -4
and when y = -4 then we get x = 4
Therefore, the closest points are:(-4,4,0) and (4, -4, 0)
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Red=10
blue=8
yellow=5
what is the ratio of red balls to blue balls?
Answer:1.25
Step-by-step explanation:
it just math
Find the surface area and width of a rectangular prism with height of 6 cm, length of 5 cm, and the
volume of 240 cm³.
Answer:
236 cm^2 and 8 cm
Step-by-step explanation:
width=w
240=6(5)(w)
w=8 cm
area=2[(6)(5)+(6)(8)+(5)(8)]
area=236 cm^2
The solid below is dilated by a scale factor of 1/2. Find the volume of the
solid created upon dilation.
24
26
10
34
Answer: 4080
Step-by-step explanation:
First you have to find the area of the triangle. 24*10 = 240. 240/2 = 120. Then you multiply the area of the triangle and multiply it by 34. 120 * 34 = 4080. This means the answer is 4080
i need help with this quick please help
Answer:
19.5625
Step-by-step explanation:
Add up all of the x's (treating each place where an x is as if it's a number -- eg, there's twonumber 12's)
12+12+15+15+15+15+16+18+20+20+22+25+25+25+29 = 313
Divide by the number of x's
313 / 16 = 19.5625
a large sample of x-y data values are analyzed and reveal a correlation coefficient of-.88. which statement is correct? group of answer choices a weak negative relationship exists. the correlation is weak because r is less than -1. if r had been .88, the correlation would have been much stronger. there is no relation. a fairly strong negative linear relationship exists. *
The correct statement is that a fairly strong negative linear relationship exists between the x and y variables.
How to find the relationship between the x and y variables of correlation coefficient?The correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where a value of -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.
In this case, the correlation coefficient is -0.88, which indicates a strong negative linear relationship between the x and y variables. This means that as the value of x increases, the value of y decreases in a predictable manner.
The negative sign of the correlation coefficient indicates that the relationship is negative, meaning that as one variable increases, the other variable tends to decrease. The absolute value of the correlation coefficient, 0.88, indicates a strong relationship, meaning that the values of the two variables are closely related and can be used to predict each other's values.
Therefore, the correct statement is that a fairly strong negative linear relationship exists between the x and y variables.
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please solve correctly my grade depends on it
Just use the pythagorean theorem to solve the hypotenuse!
(3^2)+(2^2)=x^2
9+4=13^2
[tex]\sqrt{13}[/tex] = [tex]\sqrt{x}[/tex]
[tex]13^{2}[/tex] km
Hope this helps <3
A container built for transatlantic shipping is constructed in the shape of a right
rectangular prism. Its dimensions are 4 ft by 9.5 ft by 13 ft. If the container is entirely
full and, on average, its contents weigh 0.05 pounds per cubic foot, find the total
weight of the contents. Round your answer to the nearest pound if necessary
Thus, the on average the contents weight for the transatlantic shipping is found as 24.7 pounds.
Explain about the rectangular prism:a solid, three-dimensional object with six rectangular faces.It is a prism due to its uniform cross-section along its whole length.Volume is a unit of measurement for the amount of 3-dimensional space a thing occupies. Cubic units are used to measure volume.Given dimension of rectangular prism
Length l = 4ft
width w = 9.5 ft
height h = 13 ft
Volume of rectangular prism = l*w*h
V = 4*9.5*13
V = 494 ft³
Now,
1 ft³ = 0.05 pounds
So,
weight of 494 ft³ = 494*0.05 pounds
weight of 494 ft³ = 24.7 pounds
Thus, the on average the contents weigh for the transatlantic shipping is found as 24.7 pounds.
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what is the answer of this question (please i need help)
Answer:
The answer is B ([tex]x=\frac{6}{5}[/tex])
Step-by-step explanation:
We start with creating labels for the shapes that represent what they value -at first I tried multiplying the 5x by 4 but there wasn't an answer for that.
[tex]5x+4=10[/tex]
First we just simplify,
[tex]5x (-4)=10(-4)[/tex]
[tex]5x=6[/tex]
then divide,
[tex]\frac{5x}{5} =\frac{6}{5}[/tex]
and we end up with:
[tex]x=\frac{6}{5}[/tex]
or
B
I need help please I will give brainliest to the best answer...
The value of x in the intersecting chords that extend outside circle is 5
Calculating the value of xFrom the question, we have the following parameters that can be used in our computation:
intersecting chords that extend outside circle
Using the theorem of intersecting chords, we have
4 * (x + 6 + 4) = 6 * (x - 1 + 6)
Evaluate the like terms
So, we have
4 * (x + 10) = 6 * (x + 5)
Using a graphing tool, we have
x = 5
Hence. the value of x is 5
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a random sample of n equal to 64 scores is selected from a normally distributed population with mu equal to 77 and sigma equal to 21. what is the probability that the sample mean will be less than 79? hint: this is a z-score for a sample.
The probability of the sample mean being less than 79 is 77.64%
In order to solve the given problem we have to take the help of Standard error mean
SEM = ∑/√(n)
here,
∑ = population standard deviation
n = sample size
hence, the z-score can be calculated as
z = ( x' - μ)/σ/√(n)
here,
x' = sample mean
μ = population mean
σ = population standard deviation
n = sample size
adding the values into the formula
SEM = σ / √(n)
= 21/√64
= 2.625
z = (x' - μ)/SEM
= (79-77)/2.625
= 0.76
now, using standard distribution table we find that probability of a z-score is less than 0.77 then converting it into percentage
0.77 x 100
= 77%
The probability of the sample mean being less than 79 is 77.64%
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Solve for x to make A||B.
A = x + 12
B = x + 48
X = [?]
Answer:
Step-by-step explanation:= x+48=180 ( linier pair )
= x=180-48
= x=132
= x+12=180 (liner pair)
= x=180-12
= x=168
A researcher administers a treatment to a sample from a population with a mean of m = 60. If the treatment is expected to increase scores and a one-tailed test is used to evaluate the treatment effect, then the null hypothesis would state that m ³ 60.A) TrueB) False
For the given statement after evaluating both the options the correct option is true under the condition that scores increase and null hypothesis is used to find out Treatment effect.
Here, null hypothesis clearly states that there is no significant difference is observed in comparison of sample mean and population mean.
Null hypothesis refers to statistical process which takes certain assumptions regarding two sets of different variables. In the branch of science it is used to find credibility regarding a sample data.
For the given case, the null hypothesis presents that the population mean remains unchanged (m = 60) post treatment, doesn't matter if it is greater than or equal to 60. The alternative hypothesis will be increases the mean for the treatment (m > 60).
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True. Your statement is: A researcher administers a treatment to a sample from a population with a mean of m = 60. If the treatment is expected to increase scores and a one-tailed test is used to evaluate the treatment effect, then the null hypothesis would state that m ≥ 60.
The null hypothesis typically represents no effect or no difference. In this case, the null hypothesis would state that the population mean remains unchanged (m = 60) after the treatment, not that it is greater than or equal to 60. The alternative hypothesis would be that the treatment increases the mean (m > 60).
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Someone help me i will give brainliest!
The probability that the golfer will hit at least 6 times in his next 10 attempts is A. 20 %
How to find the probability ?To estimate the probability of the golfer hitting at least 6 times in his next 10 attempts using a table of random numbers, we can perform a simulation.
Let's use the given table of random numbers to simulate 10 attempts for each trial. We can consider each pair of digits as one attempt. We will perform 10 trials and count how many times the golfer hits at least 6 times in 10 attempts.
Now count the number of trials with at least 6 hits:
Trial 2, Trial 5, and Trial 9 have at least 6 hits. That's 3 out of 10 trials.
To estimate the probability, divide the number of successful trials (at least 6 hits) by the total number of trials:
Probability = (Number of successful trials) / (Total number of trials)
Probability = 3 / 10 = 0.3
The estimated probability that the golfer will hit at least 6 times in his next 10 attempts is 30%. There is no exact match among the possible answers, but the closest one is 20%.
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PLEASE HELP AND EXPLAIN AND SHOW WORK ON HOW YOU GOT THE ANSWER I WILL MARK YOU BRAINLIEST. PLEASE EXPLAIN HOW YOU GOT THE ANSWER!!!
The terms arranged in order from smallest to biggest are: (-2)³, -√25, √11, 10, and 4² after comparing the values of the final numbers.
How to arrange the terms of numbers in ascending orderWe shall first simplify the numbers to get their final values and then compare to which is smaller as follows:
4² = 4 × 4 = 16
-√25 = -5
10 = 10
√11 = 3.3166
(-2)³ = -2 × -2 × -2 = -8
In conclusion, we have by comparing the final values of the numbers the terms arranged from smallest to the biggest as: (-2)³, -√25, √11, 10, and 4².
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Write the functions in standard form:
h(x)=2(x-3)²-9
h(x)=
p(x) = -5(x + 2)² + 15
p(x)=
Answer:
[tex]h(x)=2x^2-12x+9[/tex], [tex]p(x)=-5x^2-20x-5[/tex]
Step-by-step explanation:
To get to the standard form of a quadratic equation, we need to expand and simplify. Recall that standard form is written like so:
[tex]ax^2+bx+c[/tex]
Where a, b, and c are constants.
Let's expand and simplify h(x).
[tex]2(x-3)^2-9=\\2(x^2+9-6x)-9=\\2x^2+18-12x-9=\\2x^2+9-12x=\\2x^2-12x+9[/tex]
Thus, [tex]h(x)=2x^2-12x+9[/tex]
Let's do the same for p(x).
[tex]-5(x+2)^2+15=\\-5(x^2+4+4x)+15=\\-5x^2-20-20x+15=\\-5x^2-5-20x=\\-5x^2-20x-5[/tex]
Thus, [tex]p(x)=-5x^2-20x-5[/tex]
Help please? I just need an answer. A clear explanation earns brainliest.
the simplified form of expression is: -(x² + 2x - 2)/((x+2)*(x+4))
what is expression ?
In mathematics, an expression is a combination of numbers, variables, operators, and/or functions that represents a mathematical quantity or relationship. Expressions can be simple or complex
In the given question,
To evaluate the expression 1/(x+2) - (x+1)/(x+4), we need to find a common denominator for the two terms. The least common multiple of (x+2) and (x+4) is (x+2)(x+4).
So, we can rewrite the expression as:
(1*(x+4) - (x+1)(x+2))/((x+2)(x+4))
Expanding the brackets, we get:
(x+4 - x² - 3x - 2)/((x+2)*(x+4))
Simplifying the numerator, we get:
(-x² - 2x + 2)/((x+2)*(x+4))
Therefore, the simplified expression is:
-(x² + 2x - 2)/((x+2)*(x+4))
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Determine whether ▰ABCD with vertices A(-4,6), B(-1,7), C(0,4), and D(-3,3) is a rhombus, a rectangle, a square, or none. Select all the apply.
~a.) Rhombus
~b.) Rectangle
~c.) Square
~d.) None
The only statement that is true is b, which states that the quadrilateral is a rectangle.
What is quadrilateral?A quadrilateral is a polygon with four sides and four vertices. The sum of the interior angles of a quadrilateral is always 360 degrees. Quadrilaterals can have sides of different lengths and angles of different measures, giving rise to many different types of quadrilaterals with different properties.
According to the given informationFirst, we find the lengths of the sides of the quadrilateral:
AB = √[(7-6)² + (-1+4)²] = √10
BC = √[(4-7)² + (0-0)²] = 3
CD = √[(3-4)² + (-3+0)²] = √10
AD = √[(6-3)² + (-4+1)²] = √26
Then, we find the slopes of each pair of opposite sides:
AB: (7-6)/(−1+4) = 1/3
BC: (4-0)/(0-(-1)) = 4/1 = 4
CD: (-3-(-4))/(0-(-3)) = 1/3
AD: (6-3)/(-4-(-1)) = -1/5
Now we can analyze each statement:
a.) Rhombus
A rhombus is a quadrilateral with all sides of equal length. We found that AB = CD and AD ≠ BC, so not all sides are of equal length. Therefore, statement a is false.
b.) Rectangle
A rectangle is a quadrilateral with all angles equal to 90 degrees. We can find the slopes of adjacent sides and check if they are opposite reciprocals:
AB: 1/3
BC: 4
CD: 1/3
AD: -1/5
We can see that AB and CD have slopes of 1/3 and are opposite reciprocals, and BC and AD have slopes of 4 and -1/5, respectively, and are also opposite reciprocals. Therefore, all angles of the quadrilateral are 90 degrees. Also, since AB = CD and AD ≠ BC, the quadrilateral is a rectangle. Therefore, statement b is true.
c.) Square
A square is a special type of rectangle with all sides of equal length. We found that AB ≠ AD, so not all sides are of equal length. Therefore, statement c is false.
d.) None
We have determined that the quadrilateral is a rectangle, so it is not "none". Therefore, statement d is false.
Therefore, the only statement that is true is b, which states that the quadrilateral is a rectangle.
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Students made a craft project at camp. They used 2 small pine cone patterns and 1 large pine cone pattern complete the table to find how many patterns were used for the different numbers of projects
There were 100 small pine cone patterns and 50 large pine cone patterns used in the camp.
When 50 students constructed one craft project each using two little pine cone patterns and one giant pine cone pattern, it is the question of how many small and large pine cone patterns were utilised overall:
We can begin by figuring out how many little pine cone patterns were utilized overall to solve this.
Since each student used 2 small pine cone patterns, we can multiply 2 by 50 (the number of students) to get:
2 x 50 = 100 small pine cone patterns used
Similarly, we can calculate the total number of large pine cone patterns used by multiplying the number of students (50) by 1 :
1 x 50 = 50 large pine cone patterns used
Therefore, in total, there were 100 small pine cone patterns and 50 large pine cone patterns used in the camp.
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--The complete Question is, If a camp has 50 students and each student made one craft project using 2 small pine cone patterns and 1 large pine cone pattern, how many small and large pine cone patterns were used in total? --
19.
Solve the problem.
2
Find the critical value XR corresponding to a sample size of 5 and a confidence
level of 98%.
(1 point)
O11.143
00.297
13.277
00.484
The critical value of the chi-square distribution corresponding to a sample size of 5 and a confidence level of 98% is given as follows:
0.297 and 13.277.
How to obtain the critical value?To obtain a critical value, we need three parameters, given as follows:
Distribution.Significance level.Degrees of freedom.Then, with the parameters, the critical value is found using a calculator.
The parameters for this problem are given as follows:
Chi-square distribution.1 - 0.98 = 0.02 significance level.5 - 1 = 4 degrees of freedom.Using a chi-square distribution calculator, the critical values are given as follows:
0.297 and 13.277.
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Which expressions are equivalent to 27^4/3?
Select the three correct answers.
A. 4^3
B. (27^1/3)^4
C. 3^1/4
D. 81
D) 81 is equivalent to 27^(4/3).
The expression 27^4/3 can be simplified using the rule that (a^m)^n = a^(m*n). Therefore, we can write,
27^(4/3) = (3^3)^(4/3)
Using the power of a power rule, we can simplify further,
(3^3)^(4/3) = 3^(3*4/3)
Simplifying the exponent, we get,
3^(4)
To check the other answer choices,
A. 4^3 is not equivalent to 27^4/3.
B. (27^1/3)^4 is equivalent to 27^(4/3), which we already simplified to 3^4. Therefore, this expression is also equivalent to 3^4.
C. 3^1/4 is not equivalent to 27^4/3.
D. 81 is equivalent to 3^(4).
Therefore, the expression 27^4/3 is equivalent to 3^4, which is answer choice D) 81.
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Using the graph, determine the coordinates of the x-intercepts of the parabola.
Answer:
x = -5, x = 1
As (x, y) coordinates, the x-intercepts are (-5, 0) and (1, 0).
Step-by-step explanation:
The x-intercepts are the x-values of the points at which the curve crosses the x-axis, so when y = 0.
From inspection of the given graph, we can see that the parabola crosses the x-axis at x = -5 and x = 1.
Therefore, the x-intercepts of the parabola are:
x = -5x = 1As (x, y) coordinates, the x-intercepts are (-5, 0) and (1, 0).
Compare the numbers using <, >, or =. 0. 78 ___ 0. 708 < > =
For the given numbers, 78 < 0. 708
To compare two numbers, we need to look at their values and determine which one is larger or smaller. In this case, we have 78 and 0.708. We can start by comparing their whole number parts, which are 78 and 0, respectively. Since 78 is greater than 0, we know that 78 is a larger number.
But what about the decimal parts of these numbers? To compare them, we need to look at the place value of each digit. The first digit after the decimal point in 78 is 0, and the first digit after the decimal point in 0.708 is 7. Since 7 is greater than 0, we know that 0.708 is a larger number than 0.78 in terms of their decimal parts.
Now that we have compared the whole number parts and decimal parts separately, we can combine the results to determine the final comparison. Since 78 is larger than 0 and 0.708 is larger than 0.78 in terms of their decimal parts, we can conclude that:
78 < 0.708
We use the symbol "<" here because 78 is smaller than 0.708.
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erin is playing darts at the adventure arcade. she scores a bullseye 15% of the time, and she is about to throw 5 darts. how likely is it that she will get at least one bullseye?
the likelihood of Erin getting at least one bullseye in 5 throws is 0.5563 or 55.63%.
To calculate the likelihood of Erin getting at least one bullseye, we need to first calculate the probability of her not getting a bullseye in a single throw. Since she scores a bullseye 15% of the time, the probability of her not getting a bullseye in a single throw is 85% (100% - 15%).
Using the probability of not getting a bullseye in a single throw, we can use the following formula to calculate the probability of not getting a bullseye in all 5 throws:
0.85 x 0.85 x 0.85 x 0.85 x 0.85 = 0.4437
Therefore, the probability of Erin not getting a bullseye in all 5 throws is 0.4437 or 44.37%.
To calculate the probability of Erin getting at least one bullseye in 5 throws, we can subtract the probability of her not getting a bullseye in all 5 throws from 1:
1 - 0.4437 = 0.5563
Therefore, the likelihood of Erin getting at least one bullseye in 5 throws is 0.5563 or 55.63%.
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The probability that Erin will get at least one bullseye in her 5 throws at the adventure arcade is approximately 55.63%.
To find the probability that she will get at least one bullseye in 5 throws, we can use the complementary probability.
This means we will first find the probability of her not getting a bullseye in all 5 throws, and then subtract that from 1.
Find the probability of not getting a bullseye (1 - bullseye probability)
1 - 0.15 = 0.85
Calculate the probability of not getting a bullseye in all 5 throws
0.85^5 ≈ 0.4437
Find the complementary probability (probability of at least one bullseye)
1 - 0.4437 ≈ 0.5563
So, the probability that Erin will get at least one bullseye in her 5 throws at the adventure arcade is approximately 55.63%.
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a plane travels 600 from salt lake city, utah, to oakland, california, with a prevailing wind of 30. the return trip against the wind takes longer. find the average speed of the plane in still air.
the average speed of the plane in still air is s + 30.
Let's call the average speed of the plane in still air "s" (in miles per hour).
We can use the formula:
time = distance / speed
to find the time it takes the plane to travel from Salt Lake City to Oakland with the wind and against the wind.
With the wind:
time with wind = [tex]600 / (s + 30)[/tex]
Against the wind:
time against wind =[tex]600 / (s - 30)[/tex]
time against wind > time with wind
So we can set up an inequality:
[tex]600 / (s - 30) > 600 / (s + 30)[/tex]
Multiplying both sides by [tex](s - 30)(s + 30)[/tex], we get:
[tex]600(s + 30) > 600(s - 30)[/tex]
Expanding and simplifying, we get:
[tex]600s + 18000 > 600s - 18000[/tex]
Subtracting 600s from both sides, we get:
[tex]18000 > -18000[/tex]
This inequality is true for all values of s. In other words, there are no restrictions on the value of s that would make the return trip take longer than the trip with the wind.
Therefore, we can use the average of the two speeds (with and against the wind) to find the average speed of the plane in still air:
Average speed = [tex]2s(s + 30) / (s + 30 + s - 30)[/tex]
Simplifying, we get:
Average speed = [tex]2s(s + 30) / (2s)[/tex]
Canceling the common factor of 2s, we get:
Average speed = s + 30
We know that the distance from Salt Lake City to Oakland is 600 miles, and we can use the formula:
time = distance / speed
to find the time it takes the plane to travel this distance:
time = [tex]600 / (s + 30)[/tex]
We also know that the return trip (against the wind) takes longer, so we can set up another equation:
time return trip =[tex]600 / (s - 30)[/tex]
We can use these two equations to solve for s:
[tex]600 / (s + 30) = 600 / (s - 30)[/tex]
Cross-multiplying, we get:
[tex]600(s - 30) = 600(s + 30)[/tex]
Expanding and simplifying, we get:
[tex]600s - 18000 = 600s + 18000[/tex]
Subtracting 600s from both sides, we get:
[tex]-18000 = 18000[/tex]
This is not a valid equation, so there must be no solution.
However, we can still find the average speed of the plane in still air by using the equation we derived earlier:
Average speed = s + 30
So the average speed of the plane in still air is s + 30. We don't have a specific value for s, but we can say that the average speed is equal to the speed with the wind plus 30 (which is the speed of the wind).
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dora drove east at a constant rate of 75 kph. one hour later, tim started driving on the same road at a constant rate of 90 kph. for how long was tim driving, before he caught up to dora? a. 5 hours b. 4 hours c. 3 hours d. 2 hours
Tim was driving for 5 hours before he caught up to Dora.
The answer is (a) 5 hours.
To solve this problem, we can use the formula:
distance = rate × time
Let's denote the time Tim drove as t hours.
Since Dora started driving one hour earlier, her driving time would be (t + 1) hours.
Dora's distance: 75 kph × (t + 1)
Tim's distance: 90 kph × t
Since Tim catches up to Dora, their distances will be equal:
75(t + 1) = 90t
Now we can solve for t:
75t + 75 = 90t
75 = 15t
t = 5.
The answer is (a) 5 hours.
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