We can rewrite the equation of the circle as
[tex](x-0)^2+(y-0)^2=10^2[/tex]
so that we can be in the form
[tex](x-h)^2+(y-k)^2=r^2[/tex]
When you write the equation of a circle in this form, then the center is [tex](h,k)[/tex] and the radius is [tex]r[/tex].
So, in our case, the radius of the circle is 10.
To find the length of the bridge, we find the two points where the bridge crosses the lake (i.e. we solve the system between the equations of the line and the circle), and compute the distance between those points:
[tex]\begin{cases}y=x+2\\x^2+y^2=100\end{cases}\implies\begin{cases}y=x+2\\x^2+(x+2)^2=100\end{cases}[/tex]
Solving the second equation for x, we have
[tex]x^2+(x+2)^2=100 \iff x^2+x^2+4x+4=100\iff\\2x^2+4x-96=0 \iff x^2+2x-48=0\\\iff x=-8\ \lor\ x=6[/tex]
We use the first equation to compute the correspondent values of y:
[tex]x=-8\implies y=x+2=-6 \implies P_1 = (-8,-6)[/tex]
[tex]x=6\implies y=x+2=8 \implies P_1 = (6,8)[/tex]
Now, the distance between these two points is given by the pythagorean's theorem:
[tex]d = \sqrt{(-8+6)^2+(-6+8)^2} = \sqrt{4+4}=2\sqrt{2}[/tex]
The radius of the lake is 10 units. The length of the bridge is 14√2 units. We found this by solving the equations of the circle and the line representing the bridge.
The equation of the circular lake is given as [tex]x^2 + y^2 = 100[/tex] . This equation is of the form [tex](x -b)^2 + (y - c)^2 = r^2,[/tex] where the center of circle is at (b, c) and the radius is r. In this case, the center is at (0, 0), and the radius squared is 100. Therefore, the radius (r) is:
r = √100 = 10
So, the radius of the lake is 10 units.
The bridge is represented by the function y - x = 2, which can be rewritten as y = x + 2.
To find the points of intersection between this line and the circle, substitute y = x + 2 into the circle's equation
[tex]x^2 +[/tex][tex]y^2 = 100:[/tex]
[tex]x^2 + (x + 2)^2 = 100[/tex]
Simplify this to:
[tex]x^2 + x^2 + 4x + 4 = 100[/tex]
[tex]2x^2 + 4x + 4 = 100[/tex]
[tex]2x^2 + 4x - 96 = 0[/tex]
Divide everything by 2:
[tex]x^2 + 2x - 48 = 0[/tex]
To solve this quadratic equation, use the quadratic formula x = (-b ± √([tex]b^2[/tex] - 4ac))/(2a), where a = 1, b = 2, and c = -48:
x = (-2 ± √([tex]2^2[/tex] - 4*1*(-48)))/(2*1)
x = (-2 ± √(4 + 192))/2
x = (-2 ± √196)/2
x = (-2 ± 14)/2
This results in two solutions:
x = (12)/2 = 6
x = (-16)/2 = -8
Thus, the points of intersection are (6, 8) and (-8, -6).
Finally, to find the length of the bridge, calculate the distance between these two points using the distance formula: d = [tex]\sqrt{ ((x2 - x1)^2 + (y2 - y1)^2)[/tex]
[tex]d = \sqrt{((6 - (-8))^2 + (8 - (-6))^2)[/tex]
[tex]d = \sqrt{((6 + 8)^2 + (8 + 6)^2)[/tex]
d = [tex]\sqrt{(14^2 + 14^2)[/tex]
d = [tex]\sqrt{(196 + 196)[/tex]
[tex]d = 14\sqrt{2}[/tex]
Therefore, the length of the bridge is [tex]14\sqrt{2}[/tex] units.
What is the mean, median, mode, and range of the numbers 8,15,3,6,13,9
Hello There!
To find the mean, add up all the numbers and divide according to how many numbers there are. The mean for this set would be "9"
To find the median, place the numbers in value order and find the middle. if there is no middle, add up the two numbers in middle and divide by 2. In this set, the median would be "8.5"
To find the mode or modal value, it is best to put the numbers in order. Then count how many of each number. A number that appears most often is the mode. In this set, there would be no mode.
The range of a set of data is the difference between the highest and lowest values in the set. In this set, the range would be "12"
heres your answer in a picture
can someone show me how to do this step by step? it deals with area, and i'm not very good at math in general. thank you!!
The area of a circle when you know the radius is found using the formula Area = PI x r^2
The small circle has a radius of 7 inches.
The area of the small circle is 3.14 x 7^2 = 3.14 x 49 = 153.86 square inches.
Using the circumference of a circle use the formula Area = C^2 / 4*PI
Area of larger circle = 113.097^2 / 4 *3.14 = 12790.93141 / 12.56 = 1018.39 square inches.
Now to find the blue area subtract the area of the smaller circle from the larger one:
1018.39 - 153.86 = 864.53 square inches.
A bag contains 14 gray, 7 black and 10 white marbles. A marble is drawn at random. What is the theoretical probability, as a decimal, of drawing a black marble? Round the decimal to the nearest hundredth.
14 + 7 + 10 = 31 marbles total
7/31 = 0.23 or 23%
The probability of drawing a black marble from a bag containing 14 gray, 7 black, and 10 white marbles is 7/31, which, as a decimal rounded to the nearest hundredth, is 0.23.
The theoretical probability of drawing a black marble from a bag containing 14 gray, 7 black, and 10 white marbles can be calculated by dividing the number of black marbles by the total number of marbles in the bag.
To find the total number of marbles, we add up the counts for each color: 14 gray + 7 black + 10 white = 31 marbles in total.
Next, we calculate the probability by taking the number of black marbles and dividing it by the total number of marbles: Probability = (Number of black marbles) / (Total number of marbles) = 7 / 31.
Converting that probability into a decimal gives us approximately 0.2258. However, rounding this to the nearest hundredth, as asked in the question, gives us 0.23. So, the probability of drawing a black marble is 0.23.
CAN ANYONE HELP ME ANSWER THIS PLEASE
Answer:
3/10 * 3/10 = 9/100
Step-by-step explanation:
First chance to draw pink is 3 out of 10. replace the marble then I have another 3 out of 10 chance.
help!!!!!!!!!!!!!!!!!!!!!!!!!
Answer:
64π
Step-by-step explanation:
8²·π = 64π
ANSWER
201.06 square units.
EXPLANATION
The area of a circle is calculated using the formula,
[tex]Area = \pi \: {r}^{2} [/tex]
The given circle has radius, r=8 units.
We substitute the radius into the formula to obtain:
[tex]Area = \pi \times {8}^{2} [/tex]
The exact area is 64π square units.
When we substitute π=3.14, we get,
Area=201.06 units square.
2part question
A.predict
B what sample size
Show work
Answer:
a. 84 students
b. 12
Step-by-step explanation:
a.
From the information given, we see that 7 out of 12 practice every day. Now we want to know how many (let it equal x) would we expect to practice everyday given there are total 144 martial artists?
we set up a ratio and cross multiply and solve for x:
[tex]\frac{7}{12}=\frac{x}{144}\\7*144=12*x\\1008=12x\\x=\frac{1008}{12}=84[/tex]
So 84 students practice everyday
b.
The sample size is a "proportion" of the whole population. Here, there are a total of 144 martial artists. And we took 12 of them to figure out how many practice everyday (from these 12). So , clearly, the sample size is 12
What percent of 700 is 14 ?
Answer:
2%
Step-by-step explanation:
We are to find out what percentage of 700 is 14.
Assuming [tex] x [/tex] to be the percent of 700 which is 14, we can write it as:
[tex] \frac { x } { 100 } \times 700 = 14 [/tex]
[tex] \frac { x } {100} = \frac {14} {700} [/tex]
[tex] \frac { x } { 1 0 0 } = 0.02 [/tex]
[tex]x=0.02 \times 100[/tex]
[tex]x=2[/tex]
Therefore, 14 is 2% of 700.
What is the measure of each interior angle of a regular polygon with 5 sides?
ANSWER
108°
EXPLANATION
The measure of each interior angle of a regular polygon is given by
[tex] \theta = \frac{(n - 2) \times 180}{n} [/tex]
where n refers to the number of sides of the polygon.
From the question we have n=5 in this case.
We substitute to obtain;
[tex]\theta = \frac{(5 - 2) \times 180}{5} [/tex]
[tex]\theta = \frac{3\times 36}{1} [/tex]
This simplifies to:
[tex]\theta = 108 \degree[/tex]
Hence each interior angles of a regular polygon with 5 - sides is 108°
How much money should be invested every year with 4% interest per year in order to save up $26,000 in 18 years?
Answer:
$15116.28
Step-by-step explanation:
Given data:
interest rate, r= 4%
time, t= 18 years
Final investment, A= $26000
Principle investment, P=?
As per the formula of interest
A= P (1+rt)
Putting the values:
26000= P(1+ 0.04(18))
P= 26000/(1+ 0.04(18))
P= 15116.28 !
4x=-4r+2d for x. Mary
Final answer:
To solve the equation 4x = -4r + 2d for x, isolate x by performing the same mathematical operations on both sides of the equation.
Explanation:
To solve the equation 4x = -4r + 2d for x, we need to isolate x on one side of the equation. We can do this by performing the same mathematical operations on both sides of the equation. Let's start by moving the terms containing r and d to the other side:
4x + 4r - 2d = 0
Next, we can factor out the common factor of 4:
4(x + r - 0.5d) = 0
Finally, we divide both sides of the equation by 4 to solve for x:
x + r - 0.5d = 0
The solution for x is x = -r + 0.5d.
What is the value of X?
Answer:
I think B. 10 2/3
Step-by-step explanation:
I haven't learned this yet but out of experience i think this is how you solve it. Find out the ratio of 18 to 8 by dividing 18/8 to get 2.25. then multiply this by 24 to get 10.66 repeated which equals to 10 and 2/3.
anwser asap 50 points
What are the dimensions of the matrix?
[1 0 7 -5 9 ]
[-2 3 10 8 5]
Enter your answers in the boxes.
The matrix is a
×
matrix.
Answer:
[1, 5]
Step-by-step explanation:
Here I see ONE row and FIVE columns, so the dimensions are [1, 5].
Answer: 2,5
Step-by-step explanation:
If the dimensions of a rectangular garden can be represented by (2x+11) and (3x+5), then what is the area of the garden?
Answer: 6x^2+43x+55
Area=L*W
Area=(2x+11)(3x+5)=x^2+33x+10x+55
Area=6x^2+43x+55
Answer:
the area of the garden would be 6x^+43x+55
Lin climbed downard from 2:00 p.m. to 3:00 p.m. Add a point to the graph that shows her possible elevation at 3:00 p.m. Explain your reasioning.
To show Lin's elevation at 3:00 p.m on a graph, mark a point below her starting altitude at 2:00 p.m, given that she was climbing downwards. The exact placement will depend on her descent rate, for which we have no data.
Explanation:The question refers to Lin climbing downwards in the period from 2:00 p.m. to 3:00 p.m. In order to plot Lin's movement on a graph, we need to define the vertical axis as her altitude and the horizontal axis as time. Since Lin was moving downward, her altitude at 3:00 p.m. should be less than her altitude at 2:00 p.m.
For example, if she started at an altitude of 4,000 meters at 2:00 p.m., when adding a point to the graph at 3:00 p.m., it should be below the 4,000-meter altitude point, since Lin was descending. The exact point will depend on the speed of her descent, which we don't have information on, but it would be somewhere below the starting point.
It's similar to the thrown rock example where the rock rises and then falls. Lin, like the rock, started at a higher point (instead of airborne, she was high on a cliff or a hill), but then moved downwards, decreasing her altitude over time.
Learn more about plotting descent on a graph here:https://brainly.com/question/29164788
#SPJ12
The point is placed at -50 feet to represent a possible elevation halfway between 0 feet and -100 feet. However, it is important to note that her actual elevation could be anywhere within this range.
Assuming Lin started climbing from an elevation of 0 feet at 2:00 p.m., she could have reached a negative elevation at 3:00 p.m. This is because the graph shows that she was descending at a constant rate.
To determine her possible elevation, we need to consider the following:
The slope of the graph represents the rate of change in elevation. A negative slope indicates that Lin is descending.
The y-intercept of the graph represents Lin's elevation at 2:00 p.m. Since she started at 0 feet, the y-intercept is 0.
The time interval from 2:00 p.m. to 3:00 p.m. is one hour.
Using this information, we can determine that Lin's elevation at 3:00 p.m. could be anywhere from 0 feet to -100 feet. This is because the slope indicates that she is descending at a rate of -100 feet per hour.
Find the measure of CD. Round to the nearest tenth.
PLEASE HELP!!
The measure of arc CD is 88.8°
In circle geometry , there are certain theorems that guides the solving of problem involving circles.
Some of the theorems are ;
angle at the center is twice angle at the circumference.
The measure of arc is the measure of angle substended at the centre.
Using trigonometric ratio to get the angle at the center.
sinX = 6.35/9.06
sinX = 0.70
X = 44.4°
angle at the centre = 2 × 44.4
= 88.8°
Therefore, arc CD is 88.8°
If two angles of one triangle are congruent to two angles of another triangle, then the _____ are congruent.
Answer:
TRIANGLES
Step-by-step explanation:
The surface area of a sphere is 576pi square units. Which choice represents the volume of the sphere?
Answer:
[tex]\large\boxed{V=2304\pi}[/tex]
Step-by-step explanation:
The formula of a surface area of a sphere:
[tex]S.A.=4\pi R^2[/tex]
R - radius
We have
[tex]S.A.=576\pi[/tex]
Substitute and solve for R:
[tex]4\pi R^2=576\pi[/tex] divide both sides by π
[tex]4R^2=576[/tex] divide both sides by 4
[tex]R^2=144\to R=\sqrt{144}\\\\R=12[/tex]
The formula of a volume of a sphere:
[tex]V+\dfrac{4}{3}\pi R^3[/tex]
Substitute:
[tex]V=\dfrac{4}{3}\pi(12^3)=\dfrac{4}{3}\pi(1728)=(4)\pi(576)=2304\pi[/tex]
Which expression is equivalent for this
Answer: Option A
Step-by-step explanation:
Remember that
[tex]+ * - = -\\- * + = -\\+ * + = +\\- * - = +\\[/tex]
We have the expression
[tex]\frac{1}{2}x + (-7)-2*\frac{1}{4}-(-2)[/tex]
Solve first the multiplication of signs
[tex]\frac{1}{2}x + (-7)-2\frac{1}{4}x-(-2)\\\\\frac{1}{2}x -7-2\frac{1}{4}x+2\ \ \ \ \ \ \ \ Note:\ 2\frac{1}{4}=2+\frac{1}{4}=\frac{9}{4}\\\\\frac{1}{2}x -\frac{9}{4}x-7+2\ \ \ \ \ \ \ \ Note:\ \frac{1}{2}-\frac{9}{4}=-\frac{7}{4}=-1\frac{3}{4}\\\\-1\frac{3}{4}x -5[/tex]
Which equation can be used to represent the statement?
Sixteen is the same as one and two-tenths times a number added to negative 8.
Answer:
of the number is X, 16 = 1 + (2/10)×X - 8
Answer:
16 = [tex]\frac{6x}{5}[/tex] - 8
Step-by-step explanation:
Topic: One Unkown equation, mixed numbers
first we have to establish our variables, we only have one so we call
x = "a number"
and we start building our one unknown equation part by part by reading
"Sixteen is the same" means in math that "x ="
"one and two tenths" 1 [tex]^{2/10}[/tex]
this means we have (remember how to sum fractions?)
1 + 2 / 10 = 10/10 + 2/10 = 12/10 = 6/5
so " one and two tenths times a number added to negative 8" become [tex]\frac{6x}{5}[/tex] + (-8)
so we have our equation
16 = [tex]\frac{6x}{5}[/tex] + (-8) or 16 = [tex]\frac{6x}{5}[/tex] - 8
Refer to matrix B and identify the matrix element b23
A. -5
B. 0
C. 9
D. 8
Answer:
C. 9
Step-by-step explanation:
Matrix is an arranged array of rows and columns. The number of rows are represented by m and number of columns by n. Hence the elements of matrix is denoted by small letter with it's row number and column number in subscript.
Like elements belonging to A matrix will be written as:
[tex]a_{mn}[/tex]
In the given matrix, we have to tell the element b_23 which means we have to point out the element in row 2 and column 3.
Looking at the matric we can see that the number at the position of 2nd row and third column is 9.
Hence Option C is the correct answer ..
Two numbers total 76 and have a difference of 18. Find the two numbers
Answer:
20 and 56
Step-by-step explanation:
76/2 = 38
38 - 18 = 20
38 + 18 = 56
20 + 56 = 76
6x-3 <-9 can someone help me asap
Answer:
x<-1
Step-by-step explanation:
6x-3 <-9
Add 3 to each side
6x-3+3 <-9+3
6x < -6
Divide by 6
6x/6 < -6/6
x < -1
Answer:
x<-1
Step-by-step explanation:
6x-3<-9
Add 3
6x<-6
-6/6=-1
x<-1
(-2/7) (5/-8) please solve
Answer:
-15
Step-by-step explanation:
(-2/7) = 5
(5/-8) = -3
5 x -3 = -15
Answer:
0.17 or 5/28
Explanation:
(-2/7)=-0.28
(5/-8)=-0.62
(-0.28)(5/-8)=0.17
What is opposite sides parallel
Answer:
Opposite sides of parallel are just like diagonal and the other diagonal going the other way.
Step-by-step explanation:
What is (-6-4)divided by(-5)=
= -2? i dont really know either
Answer: 2
Step-by-step explanation:
-6-4=-10 -10/-5=2
Diep bought a baguette loaf of bread 65 centimeters long. For lunch every afternoon, he cuts 15 centimeters of bread for his
sandwich. Diep wants to determine the length of the loaf of bread, 1, after d days. What is the equation of the scenario? Is
the graph of the equation continuous or discrete?
165 - 15d, discrete
165 - 15d, continuous
65 = 1 - 15d, discrete
65 = 1 - 15d, continuous
Answer:
l = 65 – 15d; discrete
Step-by-step explanation:
15 is the centimeters he cuts off every day, you have to multiply that by D that´s the nomber of days, in order to calculate how much baguette does he have left.
If 4 days have passed, the anser would be 5 centimeters:
I=65- 15(4)= 65-60= 5
Answer:
[tex]l=65-15d[/tex]; discrete.
Step-by-step explanation:
Let d represent number of days.
We have been given that for lunch every afternoon, Diep cuts 15 centimeters of bread for his sandwich. The length of loaf cut in d days would be [tex]15d[/tex].
Diep bought a baguette loaf of bread 65 centimeters long. The length of loaf of bread after d days would be initial length minus amount of loaf cut in d days that is:
[tex]65-15d[/tex]
Since [tex]l[/tex] represents length of bread, so our equation would be: [tex]l=65-15d[/tex]
Therefore, the expression [tex]l=65-15d[/tex] represents the given scenario.
The variable d represents number of days and days cannot be fractional. We can take only whole number of days, therefore, the graph of the equation would be discrete.
What is the approximate circumference of a circle that has a center at (3,1) and passes through the point (3,6)?
Answer:
Approximately 31.42
Step-by-step explanation:
You use the Distance Formula
d= √( x2 - x1 )^2 + ( y2 - y1 )^2
substitute
r= √(3-3)^2 + (6-1)^2
√0^2 + 5^2
r = √25
r = 5
Now you can use the equation for finding the circumference
2πr
2π5
≈ 31.42
PLEASE HELP!!!!!!!
Use an equation to find the value of each indicated variable.
Answer:
y=36°
z=54°
Step-by-step explanation:
3y+2y -> 5y=180 y=36
90-36= 54
an equation that equals 19 x
Answer:
361x ÷ 19 = 19x
Answer:
18/x = 19
18=19x
Step-by-step explanation:
F(x)=square root of x^2+24x+144. g(x) = square root of x^3 -216
The square root of f(x) is ±(x + 12) and the square root of g(x) is not defined.
What is function?In mathematics, a function is an expression, rule, or law that describes the relationship between one variable (the independent variable) and another variable (the dependent variable).
f(x) = x^2 + 24x + 144
f(x) = x^2 + 2(12)x + 12^2
f(x) = (x + 12)^2
Square root of f(x) is ±(x + 12)
g(x) = x^3 - 216
g(x) = (x - 6)(x^2 + 6x + 36)
Here, (x^2 + 6x + 36) does not have real roots hence it is not possible to find the square root of g(x).
Learn more about functions on:
https://brainly.com/question/10439235
#SPJ2