The dimensions of the Norman window that admit the greatest possible amount of light are approximately:
Width (w) ≈ 11.72 ft
Height (h) ≈ 2.91 ft
To find the dimensions of the Norman window that admit the greatest possible amount of light, we need to maximize the area of the window. The window consists of a rectangle and a semicircle, so the area is the sum of the areas of both shapes.
Let's assume the width of the rectangle is "w" and the radius of the semicircle is "r".
Since the diameter of the semicircle is equal to the width of the rectangle, the radius "r" is half of "w".
Area of the rectangle = w * h, where h is the height of the rectangle.
Area of the semicircle = (1/2) * π * r²
The perimeter of the window is given as 30 ft, which can be written as:
Perimeter = 2 * (w + h) + π * r + w
Since r = w/2, we can rewrite the perimeter equation as:
Perimeter = 2 * (w + h) + (π/2) * w + w
Perimeter = 2w + 2h + (π/2 + 1) * w
Given that the perimeter is 30 ft, we have:
30 = 2w + 2h + (π/2 + 1) * w
Now, we can express "h" in terms of "w" using the perimeter equation:
h = (30 - 2w - (π/2 + 1) * w) / 2
Next, let's express the area "A" of the window in terms of "w" using the formulas for the area of the rectangle and the semicircle:
Area (A) = Area of rectangle + Area of semicircle
A = w * h + (1/2) * π * r²
A = w * ((30 - 2w - (π/2 + 1) * w) / 2) + (1/2) * π * (w/2)²
A = w * ((30 - 2w - (π/2 + 1) * w) / 2) + (1/2) * π * (w² / 4)
Now, we want to maximize the area "A."
To find the maximum value, we take the derivative of "A" with respect to "w" and set it equal to zero:
dA/dw = (30 - 2w - (π/2 + 1) * w) / 2 + (π/4) * w = 0
Solving for "w":
(30 - 2w - (π/2 + 1) * w) / 2 + (π/4) * w = 0
(30 - 2w - (π/2 + 1) * w) + (π/2) * w = 0
(30 - (2 + π/2) * w) + (π/2) * w = 0
30 - (2 + π/2) * w + (π/2) * w = 0
(30 - 2w) + (π/2 - π/4) * w = 0
30 - 2w + (π/4) * w = 0
(π/4) * w - 2w = -30
w ((π/4) - 2) = -30
w = -30 / ((π/4) - 2)
w ≈ 11.72 ft
Now that we have the value of "w," we can find the value of "h" using the perimeter equation:
Perimeter = 2w + 2h + (π/2 + 1) * w
30 = 2(11.72) + 2h + (π/2 + 1) * (11.72)
30 = 23.44 + 2h + (π/2 + 1) * 11.72
2h = 30 - 23.44 - (π/2 + 1) * 11.72
2h = 6.56 - (π/2 + 1) * 11.72
h = (6.56 - (π/2 + 1) * 11.72) / 2
h ≈ 2.91 ft
So, the dimensions of the Norman window that admit the greatest possible amount of light are approximately:
Width (w) ≈ 11.72 ft
Height (h) ≈ 2.91 ft.
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Planes A and B both intersect plane S. Vertical plane S intersects horizontal plane A and horizontal plane B. Plane S and plane A intersect at line f. Line f contains points N and K. Plane S and plane B intersect at line g. Line g contains points P and Q. Line d intersects plane A at point L. Which statements are true based on the diagram? Select three options.
Answer:
first second third
Step-by-step explanation:
Answer:
A,B,C
Step-by-step explanation:
I took the course
The mean of 100 numerical observations is 51.82 what is the value of all 100 numbers
Answer: 5182
To get the value of all 100 numbers you would need to multiply.
Step-by-step explanation:
51.82x100= 5182
[tex]\frac{5x-11}{2x^2+x-6}[/tex] You need to work for your points now!
Answer:
[tex]\frac{5x-11}{\left(2x-3\right)\left(x+2\right)}[/tex]
Step-by-step explanation:
[tex]\frac{5x-11}{2x^2+x-6}[/tex]
Factor the denominator.
[tex]\frac{5x-11}{\left(2x-3\right)\left(x+2\right)}[/tex]
The fraction cannot be simplified further.
Answer:
[tex] \frac{5x - 11}{(x + 2)(2x - 3)} [/tex]solution,
[tex] \frac{5x - 11}{2 {x}^{2} + x - 6} \\ = \frac{5x - 11}{2 {x}^{2} + (4 - 3)x - 6} \\ = \frac{5x - 11}{2 {x}^{2} + 4x - 3x - 6 } \\ = \frac{5x - 11}{2x(x + 2) - 3(x + 2)} \\ = \frac{5x - 11}{(x + 2)(2x - 3)} [/tex]
Hope this helps..
Pls help me I’ll mark brainLiest
Answer:y times 20 p
Step-by-step explanation:
Find the value of x that makes A||B
Answer:
For lines A and B to be parallel, the Same Side Interior angles must be supplementary which means:
2x + 10 + 4x + 80 = 180
6x + 90 = 180
6x = 90
x = 15°
Please answer this correctly
Answer:
0
Step-by-step explanation:
3 cards
P( odd) = 1 odd/ 3 cards = 1/3
No replacement
2 cards 6,8
No odds
P( odd) = 0/2
P( odd, no replacement, odd) = 1/2 * 0 = 0
A triangle with side lengths of 4 , 5 , 6 , what are the measures of it angles to the nearest degree ?
Answer:
41°, 56°, 83°
Step-by-step explanation:
We can find the largest angle from the law of cosines:
c² = a² +b² -2ab·cos(C)
C = arccos((a² +b² -c²)/(2ab))
C = arccos((4² +5² -6²)/(2(4)(5))) = arccos(5/40) ≈ 82.8192°
Then the second-largest angle can be found the same way:
B = arccos((4² +6² -5²)/(2·4·6)) = arccos(27/48) ≈ 55.7711°
Of course the third angle is the difference between the sum of these and 180°:
A = 180° -82.8192° -55.7711° = 41.4096°
Rounded to the nearest degree, ...
the angles of the triangle are 41°, 56°, 83°.
What tool is used to draw circles
Answer:
Pair of compasses.
Step-by-step explanation:
These are used to inscribe circles/arcs.
Compasses are used in maths, navigation,e.t.c.
Hope it helps.
a zero dimensional object that defines a specific location on a plan
Answer:
This is correct! Great Job!
Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).
Consider the given function.
Answer:
to determine the inverse of the given function, change f(x) to y, switch [tex]\boxed{x}[/tex] and y and solve for [tex]\boxed{y}[/tex]
The resulting function can be written as
[tex]f^{-1}(x)=x^2+\boxed{4}[/tex] where [tex]x\geq\boxed{0}[/tex]
Step-by-step explanation:
Hello,
f is defined for [tex]x\geq 4[/tex] as x-4 must be greater or equal to 0
and [tex]f(x)\geq 0[/tex]
so [tex]f^{-1}[/tex] is defined for [tex]x\geq 0[/tex]
and then we can write
[tex]x=(fof^{-1})(x)=f(f^{-1}(x))=\sqrt{f^{-1}(x)-4} \ so\\f^{-1}(x)-4=x^2 <=> f^{-1}(x)=x^2+4[/tex]
hope this helps
Jimmie invested $13,000 at 5.23% compounded monthly.
What will Jimmie's account balance be in 42 years?
Answer:
116370.197$
Step-by-step explanation:
Jimmie invested $13,000 at 5.23% compounded monthly.
Jimmie's account balance (B) after 42 years:
B = principal x (1 + rate)^time
= 13000 x (1 + (5.23/100)/12)^(42 x 12)
= 116370.197$
Using the diagram below, solve the right triangle. Round angle measures to the
nearest degree and segment lengths to the nearest tenth.
Answer:
m∠A = 17 degrees m∠B = 73 degrees m∠C = 90 (given) a = 12 (given) b = 40 c = 42 (given)
Step-by-step explanation:
Use sin to solve m∠A
sin x = 12/42 Simplify
sin x = 0.2857 Use the negative sin to solve for x
sin^-1 x = 17 degrees
Add together all of the angle measures to solve for m∠B
17 + 90 + x = 180 Add
107 + x = 180
-107 -107
x = 73 degrees
Use Pythagorean Theorem to solve for b
12^2 + x^2 = 42^2 Simplify
144 + x^2 = 1764
-144 -144
x^2 = 1620 Take the square root of both sides
x = 40
a bag contains only red and blue counters the probability that a counter is blue is 0.58 A counter is picked at random What is the probability that it is red
Answer:
0.42
Process:
1 - 0.58
0.42
A survey of enrollment at 35 community colleges across the United States yielded the following figures:
6414; 1550; 2109; 9350; 21828; 4300; 5944; 5722; 2825; 2044; 5481; 5200; 5853; 2750; 10012; 6357; 27000; 9414; 7681; 3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493; 2771; 2861; 1263; 7285; 28165; 5080; 11622
a. Organize the data into a chart with five intervals of equal width. Label the two columns "Enrollment" and "Frequency."
b. Construct a histogram of the data.
c. If you were to build a new community college, which piece of information would be more valuable: the mode or the mean?
d. Calculate the sample mean.
e. Calculate the sample standard deviation.
f. A school with an enrollment of 8000 would be how many standard deviations away from the mean?
Answer: (a) The chart is in the first attachment named table frequency.
(b) The histogram is in the second attachment named frequency vs. enrollment
(c) Mode
(d) x = 9071.4
(e) s = 6677.64
(f) It is -0.16 standard deviation away
Step-by-step explanation:
(c) Mode is the number in the data set which appears more often. When thinking about builiding a new community college, if you choose mode will have which college enrollment will appear more often, i.e., which courses have more students wanting to enroll.
(d) To calculate sample mean of a frequency data:
1) Find the midpoint for each interval;
2) Multiply each midpoint for its correspondent frequency;
3) Sum up each multiplication obtained in the previous step;
4) Sum up all the frequencies;
5) Divide the sum in step 3 by the sum in step 4;
For this chart:
x = [tex]\frac{3000.10+7500.16+12500.3+17500.3+22500.1+27500.2}{35}[/tex]
x = 9071.4
(e) To find the standard deviation:
1) With each midpoint, calculate its square;
2) Multiply the midppoint square by its correspondent frequency;
3) Use the following formula to determine the sample standard:
s = √∑f.M² - n(μ)² / n-1
For this chart:
s = [tex]\sqrt{\frac{4396250000 - 35*(9071.4)^{2}}{34} }[/tex]
s = 6677.64
(f) To find how many standard deviations away is the enrollment:
z = [tex]\frac{8000-9071.4}{6677.64}[/tex]
z = - 0.16
8000 enrollments are -0.16 standard deviations away from the mean.
Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.
n = 130
x = 69; 90% confidence
a. 0.463 < p < 0.599
b. 0.458 < p < 0.604
c. 0.461 < p < 0.601
d. 0.459 < p < 0.603
Answer:
d. 0.459 < p < 0.603
Step-by-step explanation:
We have to calculate a 90% confidence interval for the proportion.
The sample proportion is p=0.531.
[tex]p=X/n=69/130=0.531[/tex]
The standard error of the proportion is:
[tex]\sigma_p=\sqrt{\dfrac{p(1-p)}{n}}=\sqrt{\dfrac{0.531*0.469}{130}}\\\\\\ \sigma_p=\sqrt{0.001916}=0.044[/tex]
The critical z-value for a 90% confidence interval is z=1.645.
The margin of error (MOE) can be calculated as:
[tex]MOE=z\cdot \sigma_p=1.645 \cdot 0.044=0.072[/tex]
Then, the lower and upper bounds of the confidence interval are:
[tex]LL=p-z \cdot \sigma_p = 0.531-0.072=0.459\\\\UL=p+z \cdot \sigma_p = 0.531+0.072=0.603[/tex]
The 90% confidence interval for the population proportion is (0.459, 0.603).
what is 18576939*47 Thanks!!!!!
Answer:
873116133
Step-by-step explanation:
18576939 × 47
Multiply the numbers.
= 873116133
Answer: 873116133
Step-by-step explanation:
‘Hope this helps:)
Suppose that early in an election campaign, a telephone poll of 800 registered voters shows that 460 favor a particular candidate. Just before Election Day, a second poll shows that 520 of 1,000 registered voters now favor that candidate. At the 0.05 significance level, is there sufficient evidence that the candidate's popularity has changed?
Answer:
Yes. At the 0.05 significance level, there is enough evidence to support the claim that the proportion that support the candidate has significantly changed.
Step-by-step explanation:
This is a hypothesis test for the difference between proportions.
The claim is that the proportion that support the candidate has significantly changed.
Then, the null and alternative hypothesis are:
[tex]H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2\neq 0[/tex]
The significance level is 0.05.
The sample 1, of size n1=800 has a proportion of p1=0.58.
[tex]p_1=X_1/n_1=460/800=0.58[/tex]
The sample 2, of size n2=1000 has a proportion of p2=0.52.
[tex]p_2=X_2/n_2=520/1000=0.52[/tex]
The difference between proportions is (p1-p2)=0.05.
[tex]p_d=p_1-p_2=0.58-0.52=0.05[/tex]
The pooled proportion, needed to calculate the standard error, is:
[tex]p=\dfrac{X_1+X_2}{n_1+n_2}=\dfrac{464+520}{800+1000}=\dfrac{980}{1800}=0.54[/tex]
The estimated standard error of the difference between means is computed using the formula:
[tex]s_{p1-p2}=\sqrt{\dfrac{p(1-p)}{n_1}+\dfrac{p(1-p)}{n_2}}=\sqrt{\dfrac{0.54*0.46}{800}+\dfrac{0.54*0.46}{1000}}\\\\\\s_{p1-p2}=\sqrt{0.00031+0.000248}=\sqrt{0.000558}=0.02[/tex]
Then, we can calculate the z-statistic as:
[tex]z=\dfrac{p_d-(\pi_1-\pi_2)}{s_{p1-p2}}=\dfrac{0.05-0}{0.02}=\dfrac{0.05}{0.02}=2.33[/tex]
This test is a two-tailed test, so the P-value for this test is calculated as (using a z-table):
[tex]\text{P-value}=2\cdot P(z>2.33)=0.02[/tex]
As the P-value (0.02) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
There is enough evidence to support the claim that the proportion that support the candidate has significantly changed.
Prime factorization of 45
A. 2³×5
B. 3²×5
C. 5²×3
D. 5²×9
Answer:
Hello, your answer is:
B. 3²×5
Step-by-step explanation:
Prime factorization of 45 is:
45 = 9 x 5
= 3²×5
Hope this helps you.. Good Luck
Answer:
B. 3² × 5
Step-by-step explanation:
45 can be written as a product of its prime factors.
45 = 3 × 3 × 5
45 = 3² × 5
please help me with this
Answer:
Volume = 160 cm³ (Unit = cm³)
Step-by-step explanation:
Length = 4 cm
Width = 4 cm (Because it's a square based cuboid!)
Height = 10 cm
Now, Volume:
Volume = [tex]Length * Width*Height[/tex]
Volume = 4 * 4 * 10
Volume = 160 cm³
Answer:
160 cm³
Step-by-step explanation:
The base is a square. The side length of the base is 4 cm.
The volume of a square-based cuboid is the area of square × height or length.
4² × 10
16 × 10
= 160
The volume of the square-based cuboid is 160 cm³.
There is no overlap between the graphs of y< x+ 2 and y> x-2.
True or False
Someone help please
Answer:I think it's TRUE not sure
Step-by-step explanation:
Can somebody help me with this math question?
Answer:
8s^3
Step-by-step explanation:
In the question it says, triple the product which typically means triple using exponents. It is asking for you to do 8s x 8s x 8s or 8s to the power of 3
A ship traveled at an average rate of 25 miles per hour going west. It then traveled at an average rate of 19 miles per hour heading north. If the ship traveled a total of 145 miles in 7 hours, how many miles were traveled heading west?
Answer:
50 miles
Step-by-step explanation:
hello,
let's note x the number of miles travelled heading west,
it takes 1 hour to travel 25 miles
so it takes x/25 hours to travel x miles
we know that in total it travels 7 hours so it will travel 7-x/25 hours heading North, then heading North it takes 1 hour to travel 19 miles
so in 7-x/25 hours it travels 19(7-x/25) miles
we can write, as the total distance is 145 miles
[tex]x+19(7-\dfrac{x}{25})=145\\<=> 25x+3325-19x=3625\\<=> 6x=300\\<=> x = 50[/tex]
we can verify
50 miles heading West takes 2 hours
in 5 hours it travels 19*5 = 95 miles
the total is 145 miles
so this is correct
hope this helps
Help me plzzzzz!!!!
Answer:124
Step-by-step explanation:
2x + 8 + x - 2 = 180
Add like terms
3x + 6 = 180
Subtract the 6 from both sides
3x + 6 - 6 = 180 - 6
3x = 174
Divide by 3
x = 58
Now we have to find the measure of angle ACD
2(58) + 8 = 124
Today, the waves are crashing onto the beach every 4.8 seconds. The times from when a person arrives at the shoreline until a crashing wave is observed follows a Uniform distribution from 0 to 4.8 seconds. 61% of the time a person will wait at least how long before the wave crashes in?
Answer:
61% of the time a person will wait at least 1.872 seconds before the wave crashes in.
Step-by-step explanation:
An uniform probability is a case of probability in which each outcome is equally as likely.
For this situation, we have a lower limit of the distribution that we call a and an upper limit that we call b.
The probability that we find a value X lower than x is given by the following formula.
[tex]P(X \leq x) = \frac{x - a}{b-a}[/tex]
Uniform distribution from 0 to 4.8 seconds.
This means that [tex]a = 0, b = 4.8[/tex]
61% of the time a person will wait at least how long before the wave crashes in?
This is the 100 - 61 = 39% percentile, which is x for which [tex]P(X \leq x) = 0.39[/tex]. So
[tex]P(X \leq x) = \frac{x - a}{b-a}[/tex]
[tex]0.39 = \frac{x - 0}{4.8 - 0}[/tex]
[tex]x = 4.8*0.39[/tex]
[tex]x = 1.872[/tex]
61% of the time a person will wait at least 1.872 seconds before the wave crashes in.
Han and Clare are stuffing envelopes. Han can stuff 20 envelopes in one minute, and
Clare can stuff 10 envelopes in one minute. They start working together on a pile of
1,000 envelopes. How long does it take them to finish the pile?
Answer:
33 1/2 min
Step-by-step explanation:
One-eighth of the students in Martha's class are left handed. One-fourth of the left-handed students wear glasses. What fraction of Martha's class is left-
handed and wears glasses?
Answer:
1/32
Step-by-step explanation:
Multiply the fractions together
1/8 * 1/4 = 1/32
1/32 are left handed glass wearers
What is the solution to this inequality -13x> - 39
Answer:
Isolate the variable by dividing each side by factors that don't contain the variable.
Inequality Form: x < 3
Interval Notation:
( − ∞ , 3 )
Answer:
x<3
Step-by-step explanation:
-13x>-39
-13x>-39 (Divided by Negative Thirteen)
-13>-13
x<3 (The great sign changes to less than when divided or multiplied by a negative number.)
x={...0,1,2}
Hope this helps ❤
Sandy can fold 6 towels in 3 minutes. If she continues at this rate, how many minutes will it take her to fold 36 towels?
Hey there! :)
Answer:
x = 18 minutes.
Step-by-step explanation:
To solve this equation, set up a ratio.
# of towels over time taken:
[tex]\frac{6}{3} = \frac{36}{x}[/tex]
Cross multiply:
6x = 108
Divide both sides by 6:
6x/6 = 108/6
x = 18 minutes.
Answer:
In eighteen minutes she will have folded all 36
What is the solution to the equation below? Round your answer to two decimal places. 4+4•log2 x=4
Answer:
Option (C)
Step-by-step explanation:
Given expression is,
[tex]4+4\times \text{log}_2(x)=14[/tex]
By subtracting 4 from both the sides of the equation.
[tex]4\times \text{log}_2(x)=14-4[/tex]
Now divide the equation by 4
[tex]\text{log}_2(x)=\frac{10}{4}[/tex]
[tex]\text{log}_2(x)=2.5[/tex]
[If [tex]\text{log}_ab=x[/tex] , then [tex]b=a^{x}[/tex]]
[tex]x=(2)^{2.5}[/tex]
[tex]x = 5.657[/tex]
x ≈ 5.66
Therefore, Option C will be the correct option.
4+4•log2 x=14
x= 5.66
A surveyor is trying to find the height of a hill. He/she takes a ‘sight’ on the top of the hill and find that the angle of elevation is 40°. He/she move a distance of 150 metres on level ground directly away from the hill and takes a second ‘sight’. From this point, the angle of elevation is 22°. Find the height of the hill, correct to 1 d.p.
Answer:
The height of the hill is 116.9 meters.
Step-by-step explanation:
The diagram depicting this problem is drawn and attached below.
From Triangle ABC
[tex]\tan 22^\circ=\dfrac{h}{150+x}\\\\h=\tan 22^\circ(150+x)[/tex]
From Triangle XBC
[tex]\tan 40^\circ =\dfrac{h}{x}\\\\h=x\tan 40^\circ[/tex]
Therefore:
[tex]h=\tan 22^\circ(150+x)=x\tan 40^\circ\\150\tan 22^\circ+x\tan 22^\circ=x\tan 40^\circ\\x\tan 40^\circ-x\tan 22^\circ=150\tan 22^\circ\\x(\tan 40^\circ-\tan 22^\circ)=150\tan 22^\circ\\x=\dfrac{150\tan 22^\circ}{\tan 40^\circ-\tan 22^\circ} \\\\x=139.30[/tex]
Therefore, the height of the hill
[tex]h=139.3\times \tan 40^\circ\\=116.9$ meters( correct to 1 d.p.)[/tex]
The height of the hill is 116.9 meters.