Help 10 points and explain how you get the answer.

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

4. He drove at Puerto Princesa City to Taytay at an average speed of 40 kph

dexar [7]

Step-by-step explanation:

Distance = Speed x Time

Given Distance = 40km/h or kph

Time = 5 hours 15 minutes

=

$5 \frac{15}{60} \\ = 5 \frac{1}{4} \\ = 5.25hours$

Distance Travelled = 40 kph x 5.25 hours

= 210km

7 0

3 years ago

The area of a square is 32 cm. find the length of the diagonal

Over [174]

Answer:

8 cm

Step-by-step explanation:

Squares have equal side lengths so if we call each side x, then x × x gives us the area (32 cm²). This means that x² = 32 and therefore, one side is √32 cm.

Next, to work out the diagonal we can use Pythagoras' theory, since we can form a right-angled triangle. a² + b² = c² (c is the diagonal or the hypotenuse)

(√32)² + (√32)² = c²

(note: the square and the square root cancel out)

32 + 32 = 64

c² = 64 ∴ c = √64 which is 8

Hope this helps!

4 0

2 years ago

Read 2 more answers

Help with 16 and 17 please

Keith_Richards [23]

Answer:

16) b and 17) C

Step-by-step explanation:

8 0

3 years ago

Prove that the expression A + B - C is equivalent to the expression C - B - A if A = 2x - 1, B = 3x + 1 and C = 5x.

zepelin [54]

Answer:

both are equal is 0

Step-by-step explanation:

After substituting, the answer is 0 to both

3 0

3 years ago