What is greater: 1) 0.48 or 6/15th? 2) 0.75 or 7/8th? 3) 7.025 or 7 1/8ths?

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

An account earns simple interest. $1800 at 6.5% for 30 months Find the interest earned.

horrorfan [7]

Okay. The formula for simple interest is prt. You multiply the principal (initial amount) by the rate (percentage) by the time (months or years). $1,800 is the principal and 6.5% is the percentage rate. 1,800 * 6.5% (0.065) is 117. You earn $117 in interest annually. The time period is 30 months. There are 12 months in 1 year. Divide the amount of months by 12 to put it in a decimal. 30/12 is 2.5. Now, multiply 117 by 2.5 to find the total amount of interest earned. 117 * 2.5 is 292.5. There. The total amount of interest earned is $292.50.

4 0

3 years ago

.......................

Aliun [14]

Answer:

is something supposed to be there

Step-by-step explanation:

4 0

3 years ago

Read 2 more answers

Pepper jackie used a precision hydrometer to perform an experiment at 2 local restaurants to determine which restaurant uses the

NISA [10]

Answer:

. ALL of the beverages at The Zest Nest ARE LIKELY to contain 46 grams of sugar or less, but NONE of the beverages at Jim's juice joint ARE LIKELY to contain LESS THAN 46 grams of sugar.

3 0

3 years ago

Read 2 more answers

Find the product by using suitable properties 15 x (-25) x (-4) x (-10)

Colt1911 [192]

Answer:

−15000

Step-by-step explanation:

Solution:

15×(-25) = -375

(-4)×(-10)= -40

-375×-40= -15000

7 0

3 years ago