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2 years ago

Help this is important for my grade

Mathematics

1 answer:

andrew11 [14]2 years ago

3 0

Answer:

x+102° =180°[ Being cointerior angle]

x=180°-102°

x=78°

Also,

y=102°[ the opposite angle of parallelogram are equal]

z=78°[ the opposite of parallelogram are equal]

Hence, x=78°,y=102° and z=78°

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A norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular. Find the dimensions of a norman

Yanka [14]

Answer:

$W\approx 8.72$ and $L\approx 15.57$ .

Step-by-step explanation:

Please find the attachment.

We have been given that a norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular. The total perimeter is 38 feet.

The perimeter of the window will be equal to three sides of rectangle plus half the perimeter of circle. We can represent our given information in an equation as:

$2L+W+\frac{1}{2}(2\pi r)=38$

We can see that diameter of semicircle is W. We know that diameter is twice the radius, so we will get:

$2L+W+\frac{1}{2}(2r\pi)=38$

$2L+W+\frac{\pi}{2}W=38$

Let us find area of window equation as:

$\text{Area}=W\cdot L+\frac{1}{2}(\pi r^2)$

$\text{Area}=W\cdot L+\frac{1}{2}(\pi (\frac{W}{2})^2)$

$\text{Area}=W\cdot L+\frac{\pi}{2}(\frac{W}{2})^2)$

$\text{Area}=W\cdot L+\frac{\pi}{2}(\frac{W^2}{4})$

$\text{Area}=W\cdot L+\frac{\pi}{8}W^2$

Now, we will solve for L is terms W from perimeter equation as:

$L=38-(W+\frac{\pi }{2}W)$

Substitute this value in area equation:

$A=W\cdot (38-W-\frac{\pi }{2}W)+\frac{\pi}{8}W^2$

Since we need the area of window to maximize, so we need to optimize area equation.

$A=W\cdot (38-W-\frac{\pi }{2}W)+\frac{\pi}{8}W^2$

$A=38W-W^2-\frac{\pi }{2}W^2+\frac{\pi}{8}W^2$

Let us find derivative of area equation as:

$A'=38-2W-\frac{2\pi }{2}W+\frac{2\pi}{8}W$

$A'=38-2W-\pi W+\frac{\pi}{4}W$

$A'=38-2W-\frac{4\pi W}{4}+\frac{\pi}{4}W$

$A'=38-2W-\frac{3\pi W}{4}$

To find maxima, we will equate first derivative equal to 0 as:

$38-2W-\frac{3\pi W}{4}=0$

$-2W-\frac{3\pi W}{4}=-38$

$\frac{-8W-3\pi W}{4}=-38$

$\frac{-8W-3\pi W}{4}*4=-38*4$

$-8W-3\pi W=-152$

$8W+3\pi W=152$

$W(8+3\pi)=152$

$W=\frac{152}{8+3\pi}$

$W=8.723210$

$W\approx 8.72$

Upon substituting $W=8.723210$ in equation $L=38-(W+\frac{\pi }{2}W)$ , we will get:

$L=38-(8.723210+\frac{\pi }{2}8.723210)$

$L=38-(8.723210+\frac{8.723210\pi }{2})$

$L=38-(8.723210+\frac{27.40477245}{2})$

$L=38-(8.723210+13.70238622)$

$L=38-(22.42559622)$

$L=15.57440378$

$L\approx 15.57$

Therefore, the dimensions of the window that will maximize the area would be $W\approx 8.72$ and $L\approx 15.57$ .

8 0

3 years ago

Answer the question below. Type your response in the

lukranit [14]

<h3>Answer: 31 degrees</h3>

This is because rotations preserve angles. The angle measures won't change. That's why angle BCD is the same as angle B'C'D'. This applies to any rotation (regardless how much you rotate), any translation, any reflection, and any dilation.

Note: dilations will change the side lengths

5 0

2 years ago

The scores of 1000 students on a standardized test are normally distributed with a mean of 50 and a standard deviation of 5. wha

ra1l [238]

We are looking to find P(X>60 students)

X is normally distributed with mean 50 and standard deviation 5

We need to find the z-score of 60 students
$Z= \frac{60-50}{5}=2$

To find the probability of P(Z>2), we can do 1 - P(Z<2)
So we read the probability when Z<2 which is 0.9772, then subtract from one we get 0.0228

The number of students that has score more than 60 is 0.0228 x 1000 = 228 students

3 0

3 years ago

Hey can I have some help with this I am very confused

aniked [119]

m<5=40°

Reason-Its Given

m<2=140

Reason-Given

<5 and <2 are linear pair(looked properly in diagram) Hence they are supplementary.

<5 and 2 are same side interior angles (Clearly shown on diagram .

We can call them co responding interior angles and their sum will be 180 if lines are parallel

a||b

Let's verify

Let the angle below <1 be <3

Now

<2+<3=180

<5+<4=180

<5=<2

$\\ \sf\longmapsto$

Hence verified

4 0

3 years ago

What is n in 3/5n + 15 =10+2/5n

Phantasy [73]

Simplify both sides of the equation.

3/5n + 15 = 2/5n + 10

Subtract 2/5n from both sides.

3/5n + 15 - 2/5n = 2/5n + 10 - 2/5n15n + 15 = 10

Subtract 15 from both sides.

1/5n + 15 - 15 = 10 - 151/5n = -5

Multiply both sides by 5.

5 × (1/5n) = (5) × (−5)n = -25

Answer: n is -25.

8 0

3 years ago

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