HElp ! Its A Math Problem! How Can I solve This Math Problem

g A window is being built and the bottom is a rectangle and the top is a semi-circle. If there is 12 meters of framing materials

photoshop1234 [79]

Answer:

Semicircle of radius of 1.6803 meters

Rectangle of dimensions 3.3606m x 1.6803m

Step-by-step explanation:

Let the radius of the semicircle on the top=r

Let the height of the rectangle =h

Since the semicircle is on top of the window, the width of the rectangular portion =Diameter of the Semicircle =2r

The Perimeter of the Window

=Length of the three sides on the rectangular portion + circumference of the semicircle

$=h+h+2r+\pi r=2h+2r+\pi r=12$

The area of the window is what we want to maximize.

Area of the Window=Area of Rectangle+Area of Semicircle

$=2hr+\frac{\pi r^2}{2}$

We are trying to Maximize A subject to $2h+2r+\pi r=12$

$2h+2r+\pi r=12\\h=6-r-\frac{\pi r}{2}$

The first and second derivatives are,

Area, A(r) $=2r(6-r-\frac{\pi r}{2})+\frac{\pi r^2}{2}}=12r-2r^2-\frac{\pi r^2}{2}$

Taking the first and second derivatives

$A'\left( r \right) = 12 - r\left( {4 + \pi } \right)\\A''\left( r \right) = - 4 - \pi$

From the two derivatives above, we see that the only critical point of r

$A'\left( r \right) = 12 - r\left( {4 + \pi } \right)=0$

$r = \frac{{12}}{{4 + \pi }} = 1.6803$

Since the second derivative is a negative constant, the maximum area must occur at this point.

$h=6-1.6803-\frac{\pi X1.6803}{2}=1.6803$

So, for the maximum area the semicircle on top must have a radius of 1.6803 meters and the rectangle must have the dimensions 3.3606m x 1.6803m ( Recall, The other dimension of the window = 2r)

5 0

3 years ago

Find the missing side lengths and it's converse

Natalka [10]

Answer:

x ≈ 10.7 ft.

Step-by-step explanation:

Given triangle in the picture is a right triangle.

By applying Pythagoras Theorem in this triangle.

(Hypotenuse)² = (Leg 1)² + (Leg2)²

(14)² = x² + 9²

x² = (14)² - 9²

$x=\sqrt{196-81}$

$x=\sqrt{115}$

x = 10.72

x ≈ 10.7 ft.

8 0

3 years ago

On a bicycle ride,Betty rode 7 miles in 2 hours. At the same rate of speed,how far could she ride in 8 hours?Please hurry

prohojiy [21]

D 28 miles you want to multiply by 4 so 7x4 =28

3 0

3 years ago

What is the area of triangle ABC?

Ann [662]

Answer:

A = 3 units^2

Step-by-step explanation:

The area of a triangle is

A = 1/2 bh where b is the length of the base and h is the height

b = the distance from B to C = 4 units = 2 ( watch the scale)

h = the distance from A to C along the x axis =6 units = 3 (watch the scale)

A = 1/2 (2)(3)

A = 3 units^2

7 0

3 years ago

Read 2 more answers

A lake has a small patch of lily pads and every day the patch grows to double its size. It takes 32 days for the patch to cover

S_A_V [24]

Answer:

It took 31 days for the patch to cover half the lake

Step-by-step explanation:

The patch grows to double its size everyday

the patch completely covers the lake in 32 days

Since the patch doubles itself everyday, this means that the previous day before the 32nd day, the lake was just half covered.

Therefore, the the patch covered half the lake on the 31st day, i.e it took 31 days for the patch to cover half the lake

6 0

3 years ago