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1 year ago

F(x)=x² +6 Find an equation for the tangent line to the graph of f(x) = x² +6 at (5,31). y=

Mathematics

1 answer:

olganol [36]1 year ago

8 0

Answer:

y = x^2 + 6 given equation

y = m x + b equation of straight line required

dy/dx = 2 x tangent to given curve

dy/dx = slope = 2 * 5 = 10

Thus our line must be of the form

y = 10 x + b

31 = 10 * 5 + b at the point required

b = 31 - 50 = -19

Then our final equation becomes

y = 10 x - 19

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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

Solve for x.38(2x+16)−2=13 Enter your answer in the box.x =

lys-0071 [83]

Given:

$\frac{3}{8}(2x+16)-2=13$ $\frac{3}{8}(2x+16)=13+2$ $\frac{3}{8}(2x+16)=15$ $(2x+16)=15\times\frac{8}{3}$ $(2x+16)=40$ $2x=40-16$ $2x=24$ $x=\frac{24}{2}$ $x=12$

5 0

1 year ago

What are 3 odd consecutive integers with the sum of 3639 and what are 3 consecutive even integers with a sum of 672

Tju [1.3M]

X+(x+2)+(x+4)=3639
3x+6=3639
3x=3633
X=1211
Numbers are: 1211, 1213, 1215

X+(x+2)+(x+4)=672
3x+6=672
3x=666
X=222
Numbers are: 222,224,226

8 0

2 years ago

Whats 2.5/6 = h/9 what does h equal to

Liula [17]

Answer:

h =2.6

Step-by-step explanation:

2.5/6

2.5+6

2.6

4 0

2 years ago

Read 2 more answers

What is the answer to this picture? Thanks to anyone who helps!

Nastasia [14]

A) 3000 * 0.5 = 1500 minutes, or 25 hours.
b) 10,000 * 0.5 = 5000 minutes, or 83 hours and 20 minutes
c) she works 2 hours a weekday. 2 hours = 120 minutes. It takes her 83 hours and 20 minutes to finish a 10,000 piece puzzle. It will take her 42 weekdays to finish this. Jan 15, 2020 is a Wednesday. She will finish it on March 12, 2020.
*** I would double check my answers to make sure I didn’t make a mistake

8 0

3 years ago