A rectangle is to be inscribed in an isosceles right triangle in such a way that one vertex of the rectangle is the intersection point of the legs of the triangle and the opposite vertex lies on the hypotenuse. Find the largest area (in cm 2 ) of the rectangle and its dimensions (in cm) given that the two equal legs of the triangle have length 4.
Answers
Answer:
x = 2 cm
y = 2 cm
A(max) = 4 cm²
Step-by-step explanation: See Annex
The right isosceles triangle has two 45° angles and the right angle.
tan 45° = 1 = x / 4 - y or x = 4 - y y = 4 - x
A(r) = x* y
Area of the rectangle as a function of x
A(x) = x * ( 4 - x ) A(x) = 4*x - x²
Tacking derivatives on both sides of the equation:
A´(x) = 4 - 2*x A´(x) = 0 4 - 2*x = 0
2*x = 4
x = 2 cm
And y = 4 - 2 = 2 cm
The rectangle of maximum area result to be a square of side 2 cm
A(max) = 2*2 = 4 cm²
To find out if A(x) has a maximum in the point x = 2
We get the second derivative
A´´(x) = -2 A´´(x) < 0 then A(x) has a maximum at x = 2
The maximum area of the rectangle inscribe in the right isosceles triangle
is [tex]\rm \bold{4\;cm^2}[/tex]
Given that a rectangle is to be inscribed in an isosceles triangle
As per the attached figure we can write the following
Let ΔABC represents the isosceles triangle from the given data in the question we can write the following observations
Length of AB = 4
Length of BC = 4
∠ACB = 45°
∠BAC = 45°
Length of AC = [tex]\rm 4\sqrt2[/tex]
We have to consider a rectangle within the isosceles triangle such that
area of rectangle is maximum
The area of rectangle is given by equation (1)
[tex]\rm Area \; of \; rectangle = Length \times Breadth ........(1) \\[/tex]
Let the area of rectangle inscribed by the isosceles triangle is " A"
Let the lenght of the rectangle be "x "
Let the width of the rectangle be " y "
So according to equation (1) the area of rectangle inscribed into the triangle is given by
[tex]\rm A = x \times y.......(2)[/tex]
In ΔAED we can write
[tex]\int\limits^a_b {x} \, dx \rm tan 45= \dfrac{Perpendicular}{Base } = \dfrac{4-y}{x} \\1 = \dfrac{4-y}{x} \\ x = 4-y .......(3) \\[/tex]
From equation (2) and (3) we get
[tex]\rm A = (4-y)(y) \\A = 4y -y^2......(4)[/tex]
On differentiating the given function with respect to y we get
[tex]\rm A' = 4- 2y \\[/tex]
On putting A' =0 we get
[tex]\rm 4-2y =0 \\2y = 4\\y = 2[/tex]
A'' is negative hence it is the maximum value of y
So we can conclude that the maximum value of y is 2
So from equation (4) the maximum area is [tex]\rm 2\times 2 = 4\; cm^2[/tex]
The maximum area of the rectangle inscribe in the right isosceles triangle
is [tex]\rm \bold{4\;cm^2}[/tex]
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Related Questions
Please, I'm desperate need of help I will mark Brainliest to the one who answers with the full process! Gold will Bless You!
Two angles of a triangle are 85° and 65°, and the longest side of the triangle is 34 cm. How long is the shortest side? Include a labelled diagram.
Answers
Add 65 and 85
65+85= 150
Next subtract 180-150
The answer is 30, so that is the missing angle is 30.
The side opposite to 30 is the shortest side
Here I added the picture!
Simplify (2 1/6 - 1 2/3) - 2 2/3
Answers
Answer:
[tex]- 2\frac{1}{6}[/tex]
Step-by-step explanation:
[tex](2\frac{1}{6} -1\frac{2}{3}) - 2 \frac{2}{3}[/tex]
[tex](\frac{13}{6} -\frac{5}{3}) - 2 \frac{2}{3}[/tex]
[tex](\frac{13}{6} -\frac{10}{6}) - 2 \frac{2}{3}[/tex]
[tex]\frac{3}{6} - 2 \frac{2}{3}[/tex]
[tex]\frac{3}{6} - \frac{8}{3}[/tex]
[tex]\frac{3}{6} - \frac{16}{6}[/tex]
[tex]- \frac{13}{6}[/tex]
[tex]- 2\frac{1}{6}[/tex]
Jackson made 36 calls in 2.25 hours. How many calls per hour did he make?
Answers
Answer:
16 calls
Step-by-step explanation:
make an equation based off this ratio -- 36 calls: 2.25 hours and x calls: 1 hour
36/2.25 = x/1
cross multiply to get 36=2.25x
divided by 2.25 on both sides
x=16
In general, half of all the customers you speak to will ask about any upcoming pricing discounts or promotions. Of those who do ask, only 20% qualify for the discount. So that means if I speak to 300 customers, how many will qualify for the discount
Answers
Answer:60
Step-by-step explanation:
300 of 20% is 60
The number of qualified customers for the discount will be 60.
What is the percentage?The Percentage is defined as representing any number with respect to 100. It is denoted by the sign %. The percentage stands for "out of 100." Imagine any measurement or object being divided into 100 equal bits.
It is given that half of all the customers you speak to will ask about any upcoming pricing discounts or promotions. Of those who do ask, only 20% qualify for the discount. So that means if I speak to 300 customers.
The number of qualified customers will be calculated as below:-
The qualified customers are 20% of 300.
Number = 300 x (20/100)
Number = 300 x 0.2
Multiply the number 330 with 0.2 to get the final value.
Number = 60
Therefore, the number of qualified customers for the discount will be 60.
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Riley's cat has 8 kittens. Of those 8 kittens, 4 are brown, 1 is grey, and the rest are orange. Which of the following could be used to find the probability that the first kitten Riley picks up at random is NOT orange?
Answers
Answer:
5/8 or 62.5 percent
Step-by-step explanation:
Which equation represents the function y - 5 = –2(x+3) in slope-intercept form?
Oy= 2x + 1
Oy = 2x - 1
Oy= -2x - 1
Oy = -2x + 11
Oy = -23 - 11
Answers
Answer:
-2x-1
Step-by-step explanation:
when you graph the equation it gives you a decreasing slope with the y intercept of -1. the slope is 2 and since its decreasing it is -2. therefore its -2x-1
The solution is, the function y - 5 = –2(x+3) in slope-intercept form is y = -2x -1 .
What is a straight line?A straight line is an endless one-dimensional figure that has no width. It is a combination of endless points joined both sides of a point and has no curve.
here, we have,
given that,
the function y - 5 = –2(x+3)
now, we have to find in slope-intercept form.
now, we have,
function has a decreasing slope with the y intercept of -1. the slope is 2
and since its decreasing it is -2.
again, we have,
y - 5 = –2(x+3)
or, y = -2x - 6 + 5
or, y = -2x -1
here, slope = m = -2
and, y-intercept = c = -1
as, we know that,
slope-intercept form is: y = mx+ c
so, we get, y = -2x -1
Hence, The solution is, the function y - 5 = –2(x+3) in slope-intercept form is y = -2x -1 .
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Please answer this, I am confused (5^2)(7^2)(3^2)
Answers
Given:
The expression is:
[tex](5^2)(7^2)(3^2)[/tex]
To find:
The value of the given expression.
Solution:
We have,
[tex](5^2)(7^2)(3^2)[/tex]
On simplification, we get
[tex](5^2)(7^2)(3^2)=(25)(49)(9)[/tex]
[tex](5^2)(7^2)(3^2)=11025[/tex]
Therefore, the value of the given expression is 11025.
Identify the x-intercept points of the parabola
Answers
Answer:
(3,0) and (-1,0)
Step-by-step explanation:
The X intercept are the point(s) where the line croses the X axis (the horizontal/leftright line). the points for this graphing system are always listed (x,y) so to find a certain point you find the X value then the Y value
A university dean of students wishes to estimate the average number of hours students spend doing homework per week. The standard deviation from a previous study is 6.2 hours. How large a sample must be selected if he wants to be 99% confident of finding whether the true mean differs from the sample mean by 1.5 hours
Answers
Answer:
The Sample Size Need To be at least 114
Step-by-step explanation:
According to the Question,
Given, A university dean of students wishes to estimate the average number of hours students spend doing homework per week. The standard deviation from a previous study is 6.2 hours.The Maximum Error(E) is given as 1.5 hours and the standard deviation(σ) is 6.2 hours.
Now, For 99% Confident the level of value α is 0.01.α/2 = 0.01/2 ⇒ 0.005
Thus, From The table, We get that the value of [tex]Z_{alpha/2}[/tex] is 2.58.Hence, the sample size n can be found as
n = {[tex]\frac{(Z_{alpha/2} * Standard Deviation)}{E}[/tex]}²
n = {2.58 × 6.2}² / 1.5²
n = 113.72 .
So, The Sample Size Need To be at least 114.
0.36% as a fraction in simplest form
Answers
Answer: 9/2500
Step-by-step explanation:
We start with turning it into a fraction:
0.36/100
Then, we turn every number into a whole number by multiplying by whatever the last decimal place is called (hundreth= x100)
36/10000
Lastly, we simplify as much as we can by finding the least common denominator. In this case, the smallest whole number both 36 and 10000 can be divided by is 4:
(36/4)/(10000/4)
= 9/25000
plz it's do to day HELP PLZ
Answers
Answer:
9 units^3
Step-by-step explanation:
V = l*w*h
= 1 1/2 * 4 * 1 1/2
Change to improper fractions
= 3/2 * 4 * 3/2
= 36/4
= 9 units^3
The club with an occupancy of 100 will sell out at $25 per ticket. If the owner decides to increase the price by $5, then 2 less tickets will be sold. Determine the revenue function if x is the number of $5 increases.
Answers
Answer:
100 + (5X) = -2X
Step-by-step explanation:
Since the club with an occupancy of 100 will sell out at $ 25 per ticket, and if the owner decides to increase the price by $ 5, then 2 less tickets will be sold, to determine the revenue function if X is the number of $ 5 increases the following calculation must be performed:
100 + (5X) = -2X
So, for example, if the price were increased by $ 15, the equation would apply as follows:
15/5 = 3
100 + (5x3) = -2 x 3
100 + 15 = -6
115 = -6
There are 27 chocolates in a box, all identically shaped. They’re 10 are filled with nuts, eat with Carmel, and nine are solid chocolate. You randomly select one piece, eat it, and then select a second piece find the probability of selecting two solid chocolate in a row
Answers
There are 27 chocolates. 10 are nuts ( 37%). 9 are chocolates (33%). 8 are caremel (30%). So 37/2 = 16.5% is the answer.
11. f(x) = x + 3, g(x) = -x + 2. Find (f + g)(x)
Answers
Answer:
(f + g)(x) = 5
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightAlgebra I
Terms/CoefficientsFunctionsFunction NotationStep-by-step explanation:
Step 1: Define
Identify
f(x) = x + 3
g(x) = -x + 2
(f + g)(x) is f(x) + g(x)
Step 2: Find
Substitute in function values: (f + g)(x) = x + 3 - x + 2[Subtraction] Combine like terms (x): (f + g)(x) = 3 + 2Add: (f + g)(x) = 5Use slopes and y-intercepts to determine if the lines y=−2/3x+2 and −3x+2y=−4 are parallel.
Answers
Answer:
the answer is no
Step-by-step explanation:
y = -2/3x+2
2y = 3y-4
y = 3/2y-2
but here's a comment: it is perpendicular because it is it's opposite reciprocol
I need the area please
Answers
Answer:
120
Step-by-step explanation:
Figure shown is a parallelogram
Area of a parallelogram = base length times height
Given base length: 15in
Given height:8in
Area = 8 * 15 = 120in^2
Answer:
120 in.²
Step-by-step explanation:
8 in. × 15 in. = 120 in.²
Solve for M and N please help me
Answers
Answer:
M=4 N=6
Hope this helped!
Tell wether x and y show a proportional relationship.
x 1 2 3 4
y 4 8 12 16
Answers
Answer:
They do because y is always 4 times of x.
Step-by-step explanation:
Which expression is equivalent to 5^15 x 5^5
Answers
Answer:
[tex]5^{15} * 5^{5}[/tex]
= [tex]25^{20\\}[/tex]
An aeroplane descended 1.4 miles to an elevation of 0.6 miles as shown by the equation below.
Solve the equation to find the elevation of the plane before its descent.
x-1.4 = 0.6
A 0.8
B 1.0
C 2.0
D 1.2
Quick
Answers
Answer:
C 2.0
Step-by-step explanation:
x-1.4 =0.6
add 1.4 to both sides
x=2.0
Plz help me simply this
Answers
Answer:
Step-by-step explanation:
[tex]\sqrt[5]{u^{10}*p^{15}}*\sqrt[4]{u^{16}*p^{18}}= \sqrt[5]{u^{2*5}*p^{3*5}}*\sqrt[4]{u^{4*4}*p^{16*2}}\\\\\\=\sqrt[5]{(u^{2})^{5}*(p^{3})^{5}}*\sqrt[4]{(u^{4})^{4}*p^{4*4}*p^{2}}\\\\=\sqrt[5]{(u^{2})^{5}*(p^{3})^{5}}*\sqrt[4]{(u^{4})^{4}*(p^{4)^{4}}*p^{2}}\\\\=u^{2}*p^{3} * u^{4}*p^{4}*\sqrt[4]{p^{2}}\\\\=u^{2+4}*p^{3+4}*\sqrt[4]{p^{2}}\\\\= u^{6}p^{7}\sqrt[4]{p^{2}}[/tex]
Find the equal measures of 1. 28 m = ________ km 2. 3899 ml = ______ l 3. 79 kg = _____ g Show the process of each
Answers
Answer:
1. 0.028 km
2. 3.899 L
3. 79,000 g
WILL MARK BRAINLIEST!!
Which is the graph of g(x)
Answers
Answer:
the second one after first
Step-by-step explanation:
it strictly obeys the rules of boundary
I would appreciate if someone could answer this
Answers
Answer:
x = 15
LN and MO both equal 43
Step-by-step explanation:
LN and MO are diagonals in rectangle
Diagonals of a rectangle are congruent Hence, 4x - 17 = 2x + 13
Now we solve for x
4x - 17 = 2x + 13
Add 17 to both sides
4x = 2x + 30
Subtract 2x from both sides
2x = 30
Divide both sides by 2
x = 15
Now we want to find the values of LN and MO
To do so we substitute 15 for x in it's given expression ( note: because LN = MO we only need to do this process once )
LN = 4x - 17
Substitute 15 for x
4(15) - 17
Multiply
60 - 17
Subtract
LN and MO = 43
If using the method of completing the square to solve the quadratic equation
x^2 + 8x – 4 = 0, which number would have to be added to "complete the square"?
Answers
Step-by-step explanation:
everything can be found in the picture
Indicate the measure of each angle when ∠PTS=128
Answers
9514 1404 393
Answer:
∠PTR = 80°
∠PTQ = 45°
∠QTS = 83°
Step-by-step explanation:
The outer scale is useful for reading the values of angles that start with PT.
∡PTR = 80°
∡PTQ = 45°
The measure of angle QTS is found from ...
∡PTS = ∡PTQ +∡QTS . . . . . angle sum theorem
∡QTS = ∡PTS -∡PTQ . . . . . subtract ∡PTQ
∡QTS = 128° -45°
∡QTS = 83°
A grocery store chain uses loyalty cards that offer discounts on sale attheir store. The probability that a randomly chosen shopper has a loyaltycard is 0.8. A group of shoppers have been selected at random to completea customer satisfaction survey. Let Xi be a random variable that equals 1 when a customer has a loyalty card and 0 otherwise.
Required:
a. Construct a joint probability table for the random variables X, and X.
b. Define the event that the first four customers sampled have loyalty cards in terms of the random variables X, X2, X3, and Xd. Then compute the probability of this event.
Answers
What is half of 3/4 in fraction?
Answers
Answer:
3/8
Step-by-step explanation:
1/2 times 3/4= 3/8
What is the value of y in the equation 3y - 8 = 22?
7.3
8
10
14
Answers
Select the correct answer.
Which expression is equivalent to the glven expression?
(3m-4)^3(3m^5)
A. 81/m^2 B.27/m^7 C. 27/m^2 D.81/m^7
Answers
The correct expression which is equivalent to the given expression is,
⇒ 81 / m⁷
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The expression is,
⇒ (3m⁻⁴)³ (3m⁵)
Now, We can simplify as;
⇒ (3m⁻⁴)³ (3m⁵)
⇒ (3³m⁻¹²) (3m⁵)
⇒ (27m⁻¹²) (3m⁵)
⇒ 81 m⁻¹²⁺⁵
⇒ 81 m⁻⁷
⇒ 81 / m⁷
Thus, The correct expression which is equivalent to the given expression is,
⇒ 81 / m⁷
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