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lina2011 [118]
3 years ago
13

Find the area of the largest rectangle (with sides parallel to the coordinate axes) that can be inscribed in the region enclosed

by the graphs of f(x)=18-x^2 and
Mathematics
1 answer:
garik1379 [7]3 years ago
4 0
F(x) = 18-x^2 is a parabola having vertex at (0, 18) and opening downwards. 
g(x) = 2x^2-9 is a parabola having vertex at (0, -9) and opening upwards. 
By symmetry, let the x-coordinates of the vertices of rectangle be x and -x => its width is 2x. 
Height of the rectangle is y1 + y2, where y1 is the y-coordinate of the vertex on the parabola f and y2 is that of g.
 => Area, A 
= 2x (y1 - y2) 
= 2x (18 - x^2 - 2x^2 + 9) 
= 2x (27 - 3x^2) 
= 54x - 6x^3 
For area to be maximum, dA/dx = 0 and d²A/dx² < 0 
=> 54 - 18x^2 = 0 
=> x = √3 (note: x = - √3 gives the x-coordinate of vertex in second and third quadrants) 

d²A/dx² = - 36x < 0 for x = √3 
=> maximum area 
= 54(√3) - 6(√3)^3 
= 54√3 - 18√3 
= 36√3. 
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If 5 slices of pizza cost $5:50 how much does 
kupik [55]

The first thing we must do for this case is to define variables.

We have then:

x: number of slices

y: total cost

We write the linear function that relates the variables.

We have then:

y = (\frac{5.50}{5}) * x

Then, we evaluate the number of slices to find the total cost.

-two slices cost:

We substitute x = 2 in the given equation:

y = (\frac{5.50}{5}) * 2\\y = 2.2

Answer:

two slices = 2.2 $

-ten slices cost:

We substitute x = 10 in the given equation:

y = (\frac{5.50}{5}) * 10\\y = 11

Answer:

ten slices = 11 $

-half a slice cost:

We substitute x = 1/2 in the given equation:

y = (\frac{5.50}{5}) * \frac{1}{2}\\y = 0.55

Answer:

half a slice = 0.55 $

6 0
2 years ago
A number increased by the sum of the number and four is 20. Find the number.
miss Akunina [59]

Answer:

i think 5 is the right answer

7 0
2 years ago
Express these system specifications using the propositions p “The user enters
Alex73 [517]

Answer:

a) r ⋀~p

b)(r⋀p)⟶q

c) ~r ⟶ ~q

d) (~p ⋀r) ⟶q

Step-by-step explanation:

To solve this question we will make use of logic symbols in truth table.

We are told that;

p means "The user enters

a valid password,”

q means “Access is granted,”

r means “The user has paid the

subscription fee”

A) The user has paid the subscription fee, but does not enter a valid

password.”

Fist part of the statement is correct and so it will be "r". Second part of the statement is a negation and will be denoted by ~p. Since both statements are joined together in conjunction, we will use the conjuction symbol in between them which is "⋀" Thus, we have; r ⋀~p

B) Still using logic symbols, we have;

(r⋀p)⟶q

⟶ means q is true when r and p are true.

C) correct symbol is ~r ⟶ ~q

Since both statements are negation of the question. And also, if ~r is true then ~q is also true.

D) Similar to answer A to C above, applying similar conditions, we have (~p ⋀r) ⟶q

8 0
3 years ago
Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gen
Aleksandr-060686 [28]

Let X be the number of boys in n selected births. Let p be the probability of getting baby boy on selected birth.

Here n=10. Also the male and female births are equally likely it means chance of baby boy or girl is 1/2

P(Boy) = P(girl) =0.5

p =0.5

From given information we have n =10 fixed number of trials, p is probability of success which is constant for each trial . And each trial is independent of each other.

So X follows Binomial distribution with n=10 and p=0.5

The probability function of Binomial distribution for k number of success, x=k is given as

P(X=k) = (10Ck) 0.5^{k} (1-0.5)^{10-k}

We have to find probability of getting 8 boys in n=10 births

P(X=8) = (10C8) 0.5^{8} (1-0.5)^{10-8}

= 45 * 0.0039 * 0.25

P(X = 8) = 0.0438

The probability of getting exactly 8 boys in selected 10 births is 0.044

8 0
3 years ago
How would I find the integral of <img src="https://tex.z-dn.net/?f=%5Cint%5Cfrac%7Btdt%7D%7Bt%5E4%2B2%7D" id="TexFormula1" title
kotegsom [21]
Let t=\sqrt y, so that t^2=y, t^4=y^2, and \mathrm dt=\dfrac{\mathrm dy}{2\sqrt y}. Then

\displaystyle\int\frac t{t^4+2}\,\mathrm dt=\int\frac{\sqrt y}{2\sqrt y(y^2+2)}\,\mathrm dy=\frac12\int\frac{\mathrm dy}{y^2+2}

Now let y=\sqrt2\tan z, so that \mathrm dy=\sqrt2\sec^2z\,\mathrm dz. Then

\displaystyle\frac12\int\frac{\mathrm dy}{y^2+2}=\frac12\int\frac{\sqrt2\sec^2z}{(\sqrt2\tan z)+2}\,\mathrm dz=\frac{\sqrt2}4\int\frac{\sec^2z}{\tan^2z+1}\,\mathrm dz=\frac1{2\sqrt2}\int\mathrm dz=\dfrac1{2\sqrt2}z+C

Transform back to y to get

\dfrac1{2\sqrt2}\arctan\left(\dfrac y{\sqrt2}\right)+C

and again to get back a result in terms of t.

\dfrac1{2\sqrt2}\arctan\left(\dfrac{t^2}{\sqrt2}\right)+C
3 0
3 years ago
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