A rectangle is placed inside a circle whose diameter is 13 with all of its corners on the circle. The area of the rectangle is 60. What are the dimensions of the rectangle? Hint: Draw a picture and use the Pythagorean Theorem.
Answers
Given
A rectangle is placed inside a circle whose diameter is 13unit all corners of it on the circle.
area of the rectangle = 60 square unit
Find the dimensions of the rectangle.
To proof
As given in the question
rectangle is placed inside a circle whose diameter is 13unit .
now by using the diagram given below
the diagonal of the rectangle is equal to the diameter of the circle.
Thus DB = 13 unit
FORMULA
Area of the rectangle = Length × Breadth
area of the rectangle = 60 square unit
let the value of the length be ( Say AB ) = x
put the value in the above equation
we get
60 = x × Breadth
[tex]Breadth( say AD)= \frac{60}{x}[/tex]
now in ABD
Using the pythagorus theorem
we get
BD²= AB² + AD²
[tex]13^{2} = x^{2}+ (\frac{60}{x}) ^{2}[/tex]
solving the above equation
we get
[tex]169x^{2}= x^{4} + 3600[/tex]
[tex]x^{4} - 169x^{2}+3600 =0[/tex]
[tex]x^{4} - 144x^{2} -25x^{2} + 3600 =0[/tex]
[tex]x^{2} (x^{2} - 144) - 25 (x^{2} - 144) =0[/tex]
(x² - 25) (x² -144) =0
The roots are
x = -12 ,-5, 5, 12
-12 and -5 are neglected because of negative terms .
( Length and breadth of recangle cannot be negative )
thus x = 12 and 5
Two case arise
First case
when
Length of rectangle= 12unit
[tex]Breadth = \frac{60}{12}[/tex]
Breadth of rectangle = 5 unit
Second case
Length of rectangle= 5unit
[tex]Breadth = \frac{60}{5}[/tex]
Breadth of rectangle = 12 unit
Hence proved
Final answer:
To find the dimensions of a rectangle inside a circle with a diameter of 13, we use the Pythagorean Theorem and the area of the rectangle. After substituting values and solving, we find the dimensions are 5 units by 12 units.
Explanation:
To find the dimensions of the rectangle that fits inside a circle with a diameter of 13, one must first recognize that the rectangle forms the sides of a right-angled triangle where the diagonal of the rectangle serves as the hypotenuse of the triangle. This hypotenuse, thus, equals the diameter of the circle, which is 13 units.
Since the area of the rectangle is 60 square units, we let the sides of the rectangle be denoted as length (l) and width (w), with the area equation being l×w = 60.
Using the Pythagorean Theorem, we have: l2 + w2 = 132. We also know that lw = 60, which allows us to solve for one variable in terms of the other. Let's solve for w: w = 60/l. Substituting this into the Pythagorean equation gives l2 + (60/l)2 = 169. Solving this quadratic equation will give us the dimensions of the rectangle that can fit within the circle.
After solving, one finds that the dimensions that satisfy both the area and the Pythagorean Theorem are 5 and 12.
Therefore, the dimensions of the rectangle are 5 units by 12 units.
Related Questions
what is the mean of 5,10,15,20,25
Answers
5+10+15+20+25=75
75/5=15
15 is the mean
5+10+15+20+25=75
Your mean will be 75
Then divide by 5 = 15
Your answer will be 15
Find the difference between x+5 and 2x+3
Answers
[tex](2x + 3) - (x + 5) = 2x + 3 - x - 5 =x - 2 [/tex]
Answer:
The difference between the given expressions would be [tex]x - 2[/tex]
Step-by-step explanation:
Given phrase,
Difference between x+5 and 2x+3,
[tex]( 2x + 3 ) - (x + 5)[/tex]
[tex]=2x + 3 - x - 5[/tex] ( By distributive property )
[tex]=x - 2[/tex] ( By combining like terms )
Hence, the required difference would be [tex]x-2[/tex]
What’s is 38 divided by 100
Answers
38 divided by 100
what you could do is that 38 in decimal form will be 38.00
So,...
38.00 divided by 100
when you divided by 10 or 100 or 1,000 or etc.
all you have to do is move the decimal from 38.00 to the left 2 times
(because there are only 2 zeros in 100)
and your answer will be 0.38
38 divided by 100 = 0.38
hope that helps
D-I are the measurements of the lengths of the sides of triangles. Identify the types of triangles.
d. 45 cm 61 cm 60 cm
e. 30 cm, 31cm, 30 cm
f. 4 cm, 5ft, 4 cm
g. 89mm, 89mm, 89mm
h. 9 in, 5in, 3 in
i. 22cm, 22cm, 22in
Answers
The types of triangles are:
d. Scalene
e. Isosceles
f. Isosceles
g. Equilateral
h. Scalene
i. Equilateral
Here's the type of triangle for each set of measurements:
d. 45 cm, 61 cm, 60 cm:
All sides are different lengths.
Therefore, this is a scalene triangle
e. 30 cm, 31cm, 30 cm:
Two sides have equal lengths (30 cm).
This is an isosceles triangle
f. 4 cm, 5ft, 4 cm:
We need to convert feet to the same unit as cm for comparison.
1ft = 12 inches = 2.54 cm
5ft = 5 * 2.54 cm ≈ 12.7 cm
We have two sides with equal length (4 cm) and one different length (12.7 cm).
This is also an isosceles triangle
g. 89mm, 89mm, 89mm:
All three sides have the same length.
This is an equilateral triangle, a special type of isosceles triangle where all sides are equal.
h. 9 in, 5in, 3 in:
All sides have different lengths.
This is a scalene triangle
i. 22cm, 22cm, 22in:
We need to convert inches to centimeters for comparison.
22in = 22 * 2.54 cm ≈ 55.88 cm
Comparing the converted lengths, we see all sides have equal lengths.
This is an equilateral triangle
you are charged $16.05 after tax for a meal. Assume sales tax is 7%, what was the menu price for the meal?
Answers
[tex]1.07x=16.05[/tex]
[tex]16.05/1.07=15[/tex]
[tex]15=x[/tex]
The menu price for the meal is $15.
Write An equation in slope intercept form for the line passing through each pair of points (-1,3) and (2,-5)
Answers
Slope m = y2-y1/x2-x1
Points (-1,3) and (2,-5)
M= -5-3/2-(-1)
M= -8/3
M= -8/3
Slope is -8/3
Now fill in the formula equation of a line using one if the points and slope to find b .
(2,-5) and slope -8/3
Y=mx+b
-5=-8/3(2)+b
-5=-16/3+b
-5+16/3=b
1/3=b
Now put all together
Y= -8/3x+1/3
Slope intercept form of a line is y = mx + b, where m is the slope and b is the y coordinate of the y intercept.
The easiest way to solve this problem is by using another equation for a line, also known as point slope form. This equation is best to use for a line if you have a point and slope of the line, but do not have the y-intercept.
We need to find the slope of this line passing through the pair of points (-1, 3) and (2, -5). We can do this by using the slope formula: [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]. Substitute in the appropriate coordinates.
[tex]\frac{-5-3}{2-(-1)}[/tex] = [tex]\frac{-8}{3}[/tex], so the slope of the line is -8/3.
Point slope form of a line is [tex]y-y_1=m(x-x_1)[/tex]. Substitute -8/3 for m, and use the point coordinate (-1, 3) for y1 and x1. You could use either point coordinate but I chose to use this one.
[tex]y-3= -8/3[x-(-1)][/tex]
Distribute -8/3 inside the parentheses.
[tex]y-3= -8/3x - 8/3[/tex]
Add 3 to both sides.
y = -8/3x + 1/3 is your answer (slope intercept form).
Hello can you help me please?)
Answers
The "vertical line test" tells you the relation is not a function if any vertical line intersects the graph in 2 or more places.
15. not a function
The vertical line x=0 crosses the graph at y=-6 and at y=6.
16. a function
The midpoint of AB is (3,7). The coordinate of one endpoint is A(5,1). Find the coordinate of endpoint B. Show or explain work.
Answers
The formula of a midpoint:
[tex]M_{AB}\left(\dfrac{x_A+x_B}{2},\ \dfrac{y_A+y_B}{2}\right)[/tex]
We have
[tex]M(3,\ 7)\to x_M=3,\ y_M=7\\\\A(5,\ 1)\to x_A=3,\ y_A=7[/tex]
Substitute:
[tex]\dfrac{5+x_B}{2}=3\qquad|\cdot2\\\\5+x_B=6\qquad|-5\\\\x_B=1\\\\\dfrac{1+y_B}{2}=7\qquad|\cdot2\\\\1+y_B=14\qquad|-1\\\\y_B=13[/tex]
Answer: B(1, 13)A(5,1) which becomes (x1,y1)
P(3,7) which becomes (x,y)
B(x,y) which becomes (x2,y2)
Since it is a midpoint, we can use the midpoint formula directly.
x = (x1+x2)/2
3 = (5+x)/2
x=1
Thus our x coordinate is 1
y = (y1+y2)/2
7 = (1+y)/2
y=13
Thus our y coordinate is 13
Our answer is B(1,13)
In a recent year the total sales at a restaurant a bar $350,000 worth in at restaurant be. Total sales at the two stores were $575 million. What were the sales for the year at each store?
Answers
Subtract the difference from the total:
575,000,000 - 350,000 = 574, 650, 000
Divide the difference by 2:
574, 650, 000 / 2 = 287,325,000
That is the lower one, then add the difference for the higher one:
Restaurant A = $287,325,000 + $350,000 = $287, 675,000
Restaurant B = $287,325,000
f(x)=2^x evaluate f(5)
Answers
Hey there!
To solve this, we will first replace x with 5 anytime we see it in the equation.
f(5) = 2^5
Now, we simplify 2^5.
The base is 2 and the exponent is 5 in this case.
The exponent is the number of times you multiply the base by itself. In oour case:
2 x 2 x 2 x 2 x 2 = 32
f(5) = 32
Hope this helps!
A package weighs 2.6 pounds how much do 8 of the same sized packages weigh
Answers
you would multiply 2.6 x 8 and that equals 20.8
8 packages weigh 20.8 pounds.
Explanation: It's simple. 2.6 x 8 = 20.8
erica takes a cab to meet her grandmother at a musuem. her grandmother wiill giive her a ride hime. she only has$25.05 to spend and a cab cost.$3 plus $0.35 per mile. what are rhe maximum number if miles that Erica may travel before she has to walk the rest of the way to the museum?
Answers
The maximum miles that she can travel 63 miles before she can walk.
Final answer:
By setting up an equation that accounts for the cab's base fare and the additional per mile charge, we can determine that Erica can afford to travel a maximum of 63 miles in the cab before exceeding her budget of $25.05.
Explanation:
To determine the maximum number of miles Erica can travel in a cab without exceeding her budget of $25.05, we can create an equation where the cost of the cab is equal to the base fare plus the per mile charge times the number of miles (mileage). We know the base fare is $3 and the per mile charge is $0.35. Let's denote the number of miles Erica can travel as 'm'.
The cost equation would be: Cost = Base Fare + (Per Mile Charge × Mileage)
This gives us: $25.05 = $3 + ($0.35 × m)
Subtracting the base fare from both sides gives us: $22.05 = $0.35 × m
Dividing by the per mile charge yields: m = $22.05 / $0.35
Calculating this, we get: m = 63 miles
Therefore, Erica can travel a maximum of 63 miles in the cab before she would need to walk the rest of the way to the museum to stay within her budget.
The scale says a patient weighs 70kg.this is the same as weight how many pounds
Answers
Answer:
154 LBS
Step-by-step explanation:
1 KG = 2.2 LBS
70 KG X 2.2 = 154
To convert kilograms to pounds, you divide the weight in kilograms by 0.4536. So, a person weighing 70 kg would weigh approximately 154.32 lbs.
Explanation:The subject you're asking about is conversion from kilograms to pounds. A weight of 1 lb is equivalent on Earth to a mass of 0.4536 kg. So, to convert kilograms to pounds, you would divide the weight in kilograms by the equivalent mass of 1 pound in kilograms.
For example, a person weighing 70 kg would weigh approximately 154.32 pounds. This was calculated by dividing 70 kg by 0.4536 kg per pound (70/0.4536 = 154.32 pounds). It's important to note that while weight can vary slightly depending on location due to differences in gravitational pull, this conversion factor still provides a good estimation for most common situations.
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Which line is a graph of the equation: 2x + 5y = –10? Number graph ranging from negative five to five on the x axis and negative six to two on the y axis. Four lines are drawn in blue on the graph. Lines a and c have a positive slope and are parallel to each other. Lines b and d have a negative slope and are parallel to each other. Lines a and b intersect at (zero, two), and lines c and d intersect at (zero, negative two). A. line a B. line b C. line c D. line d
Answers
First we find the slope of the given line by using slope intercept form.
Converting the equation into slope intercept form
[tex] 5y=-2x-10[/tex]
Divide both side by 5 we get
[tex] y==-\frac{2}{5}-2[/tex]
On comparing with [tex] y=mx+c[/tex]
We get slope [tex] m=-\frac{2}{5}[/tex]
as the slope of line is negative therefore the line will either "b" or "d"
As line "b" pass from (0,2)
On putting it in given equation we get
[tex] 2(0)+5(2)=10[/tex]
Which is not equL to RHS of the line
Therefore b is not the given line
Now putting point (0,-2)
We get [tex] 0+5(-2)=-10[/tex]
Satisfy the equation and line "d" pass from it
There the give equationis of line "d"
Answer:the answer is line D
Step-by-step explanation:took the test and got it right
2x+5y=35 x=-10-15y. what value of x satisfies the system of equations?
Answers
A model car travels 20 meters in 15 seconds. How many meters will it travel in 45 seconds?
Answers
The model car will travel 60 meters in 45 seconds
Miguel baked 5 loaves of bread. Each loaf required 7/8 cup of flour. How much flour dis miquel use?
Answers
Simplify 3m^2(10+m) PLEASE HELP
Answers
3m^2 (10+m)
Use FOIL
3m^2*m=3m^3
3m^2*10= 30m^2
Answer: 3m^3 + 10m^2
Solution :
Given expression [tex]3m^{2}(10+m)[/tex]
To simplify this expression, first we need to know about the Distributive property.
Distributive Property: [tex]a\times(b+c)=(a\times b)+(a\times c)[/tex]
Given expression [tex]3m^{2}(10+m)[/tex]
Applying Distributive property on this expression -
[tex]\Rightarrow 3m^{2}(10+m)=(3m^{2}\times10)+(3m^{2}\times m)[/tex]
[tex]= 30m^{2}+3m^{3}[/tex]
Simplified form of [tex]3m^{2}(10+m)[/tex] is [tex]30m^{2}+3m^{3}[/tex]
round to the nearest hundredth 3.2649
Answers
The answer is 3.26 :D :D :D :D
The answer to rounding the number 3.2649 to the nearest hundredth is 3.26. You need to consider the digit in the thousandths place to determine if you round up or down.
Explanation:To round the number 3.2649 to the nearest hundredth, you will need to look at the digit in the thousandths place, which is 4 in this case. In rounding, if the digit is 5 or more, you round up, else you round down. Here it is 4, it's less than 5, so we round down. So, the number 3.2649 when rounded to the nearest hundredth is 3.26.
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If ƒ( x )= 2 x 2 + 3, then ƒ( x - 1)= 2 x 2 + 4 x + 5.
True or False
Answers
The value of function f(x - 1) = 2x² + 4x + 5 is not true.
Here,
The function is given as;
f (x) = 2x² + 3
And, f(x - 1) = 2x² + 4x + 5
We have to check the value of f(x - 1) is true or not.
What is substitution method?
To find the value of any one of the variables from one equation in terms of the other variable is called the substitution method.
Now,
The function is;
f (x) = 2x² + 3
To determine the value of f (x - 1), we can put (x -1) in place of x as;
f (x) = 2x² + 3
f (x - 1) = 2(x - 1)² + 3
f (x - 1) = 2(x² + 1 - 2x) + 3
f (x - 1) = 2x² + 2 - 4x + 3
f (x - 1) = 2x² - 4x + 5
Hence, The value of function f(x - 1) = 2x² + 4x + 5 is not true.
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The number of employees of a company decreased from 225 to 189. Find the percent decrease
Answers
The number of employees at the company decreased by 16%. This was calculated by first finding the decrease in number of employees (36) and then calculating this as a percentage of the original number of employees.
Explanation:To find the percent decrease in the number of employees at the company, you need to follow these steps:
First determine how many employees there has been a decrease by subtracting the final number of employees from the original number. So 225 - 189 = 36.Next, calculate the percentage that this decrease represents out of the original number. You do this by dividing the decrease by the original number and then multiplying the result by 100 to get a percentage. So (36/225) * 100 = 16%.So the number of employees at the company decreased by 16%.
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HELP PLEASE!!!!!!!! I GIVE 21 POINTS!!!!!!!!! AND BRAINLEAST <3
Answers
6. domain=age
range= number of pets
it is not a function because an x value repeats
7. domain=1,2,3,4,5
range= 50, 100, 150, 150, 200
it is a function because the lines don't connect
which fraction is not equivalent 30\48
Answers
The answer is C. 20/38
Point A has coordinate A(3, 2). The point is rotated 180° clockwise around the origin. Find the coordinates of point A’. A’(2, 3) A’(–3, –2) A’(–2, –3) A’(–2, 3)
Answers
Answer:
A' (-3,-2)
Step-by-step explanation:
When rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. To rotate a figure 180 degrees, you apply the rule (x, y) → (-x, -y). Start by using a coordinate grid with coordinates for each vertex of the figure. The center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. Write down the original coordinates of the shape you are going to rotate. Then, apply the rule. For example, coordinate (3, 2) becomes (-3, -2). Another example would be, coordinate (-4, -2) becoming (4, 2).
(-3,-2) this is your answer
The number a times 10 to the power of negative 2 is about 8 times greater than b times 10 to the power of negative 3. Find the correct value of a and b.
A. a = 3.1 and b = 2.5
B. a = 2.5 and b = 3.1
C. a = 2 and b = 1.2
D. a = 1.2 and b = 2
Answers
Answer:
c
Step-by-step explanation:
The correct values of a and b are a = 3.1 and b = 2.5, which corresponds to option A.
To solve the question, we need to set up an equation using the information given. Here's a step-by-step explanation:
a * 10^-2 = 8 * b * 10^-3Divide both sides by 10^-3 to simplify the equation: a * 10 = 8 * bPlugging in the values for a and b from option A, we get 3.1 * 10 = 8 * 2.5, which is true.Therefore, a = 3.1 and b = 2.5, matching option A.1. If ΔMNO≅ΔGHI, then which of the following is NOT true? A. ΔMON≅ΔGIH B. OM=IG C. ΔHIG≅ΔNMO D. ΔIHG≅ΔONM E. ∠GHI≅∠ONM
Answers
0.86 divide 5 -0.3 multiply 0.5
Answers
0.68÷5-0.3×0.5? or
0.86÷5 and -0.3×0.5
0.172 , -0.15
Answer:
The solution is 0.022.
Step-by-step explanation:
Given that,
[tex]\dfrac{0.86}{5}-0.3\times0.5[/tex]
We need to solve the equation
Using PEMDAS rule
On dividing 0.86 by 5
[tex]0.172-0.3\times0.5[/tex]
On multiply 0.3 from 0.5
[tex]0.172-0.15[/tex]
On subtract 0.15 from 0.172
[tex]0.022[/tex]
Hence, The solution is 0.022.
Rafael has been given a list of 4 bands and asked to place a vote . His vote must have the names of his favorite, second favorite and third favorite bands from the list. How many different votes are possible
Answers
There will be twelve votes possible because 3*4=12
[tex]4p3[/tex]
4 permutation ( because in this scenario order matters ) 3 meaning that there are 4 choices and you can pick 3 In any order
Benito saves $6 out of every $20 he earns. He earned $90 one month. How much did he save?
Answers
Set up a proportion:
6/20 = x/90
Meaning 6 saved over 20 and x saved over 90
Cross multiply
540 = 20x
x = 27
He saved $27
And 4.5 x 6 = 27
Which of the following functions is the inverse of f (x) = 3x? A. F (x) = -3x. B. F (x) = X3. C. F (x)=x/3 D. This function does not have an inverse.
Answers
[tex]f(x)=3x\to y=3x\\\\\text{Interchange the y and the x}\\\\x=3y\\\\\text{Solve for y}\\\\3y=x\qquad|:3\\\\y=\dfrac{x}{3}\\\\Answer:\ C.\ f^{-1}(x)=\dfrac{x}{3}[/tex]
The required inverse function would be f⁻¹(x) = x / 3, which is the correct answer would be option (C).
What is an inverse function?The inverse function is defined as a function obtained by reversing the given function.
To determine the inverse of a function, all you have to do is switch where x and y are and resolve for y.
We have to find the inverse of the function below :
f(x) = 3x
So after switching x and y,
Now, we solve for y regularly.
f(x) = x = 3y
Solve for y :
⇒ x = 3y
⇒ x / 3 = y
⇒ y = x / 3 = f⁻¹(x)
Therefore, the required inverse function would be f⁻¹(x) = x / 3.
Hence, the correct answer would be an option (C).
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