Find the surface area of the part of the paraboloid z=5-3x^2-2y^2 located above the xy plane. (10 points) z

Step-by-step explanation:

a) Give two pairs which are in the relation [tex]\equiv \mod 4[/tex] and two pairs that are not.

As stated before, a pair [tex](x,y)\in \mathbb{Z}\times\mathbb{Z}[/tex] is equal mod m (written [tex]x\equiv y\mod m[/tex]) if [tex]m\mid (x-y)[/tex]. Then:

b) Show the [tex]\equiv \mod m[/tex] is an equivalence relation.

An equivalence relation is a binary relation that is reflexive, symmetric and  transitive.

By definition [tex]\equiv \mod m[/tex] is a binary relation. Observe that:

[tex]m\mid [(y-x)+(z-y)] \implies m\mid (z-x) \implies x\equiv z \mod m[/tex].

In conclusion, [tex]\equiv \mod m[/tex] defines an equivalence relation.

Answer:

Step-by-step explanation:

The question is:

The amount of red blood cells in a blood sample is equal to the total amount in the sample minus the amount of plasma. What is the total amount of blood drawn? Red blood cells = 45% Plasma = 5.5 ml

Solution:

We have given:

The amount of red blood cells in a blood sample is equal to the total amount in the sample minus the amount of plasma. We need to find how many ml is 1% of blood.

The equation we get is:

Red blood cells = total sample - amount of plasma

45% = 100% - 5.5 ml

Combine the percentages:

5.5 ml = 100%-45%

5.5 ml = 55%

0.1 ml = 1%

1% = 0.1 ml

Now you can find out red blood cells volume or total sample volume.

Red blood cells volume = 45% x 0.1ml/%

= 4.5ml

Total sample volume =100% x 0.1ml/%

= 10ml .

Answer:

Step-by-step explanation:

We can work with these values as a set value, and build a Venn Diagram from them.

I am going to say the set A are those that have the health insurance plan.

Set B are those that have the life insurance plan

Set C are those that have the investment plan.

We have that:

[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]

In which a is the number of employees that only have the health insurance plan, [tex]A \cap B[/tex] is the number of employees that have both the health and the life insurance plans, [tex]A \cap C[/tex] is the number of employees that have both the health insurance and the investment plans. and [tex]A \cap B \cap C[/tex] is the number of employees that have all three of those plans.

By the same logic, we have that:

[tex]B = b + (A \cap B) + (B \cap C) + (A \cap B \cap C)[/tex]

[tex]C = c + (B \cap C) + (A \cap C) + (A \cap B \cap C)[/tex]

The problem states that:

An employee cannot have both life insurance and investment plans. So:

[tex]B \cap C = 0, A \cap B \cap C = 0[/tex]

45 employees have only the life insurance plan. So:

[tex]b = 45[/tex]

There are 20 more employees that have both health and life plans than those that have both health and investment plans

[tex]A \cap B = A \cap C + 20[/tex]

320 employees have the health insurance plan.

[tex]A = 320[/tex]

450 employees have at least one plan

[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 450[/tex]

330 employees have only one plan

[tex]a + b + c = 330[/tex]

How many people have the investment plan?

We have to find the value of C.

Now we solve:

[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 450[/tex]

Applying what we have

-----------

[tex]a + b + c + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 450[/tex]

[tex]330 + A \cap C + 20 + A \cap C = 450[/tex]

[tex]2(A\capC) = 100[/tex]

[tex]A \cap C = 50[/tex]

[tex]A \cap B = A \cap C + 20 = 50 + 20 = 70[/tex]

----------------

[tex]A = 320[/tex]

[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]

[tex]a + 70 + 50 = 320[/tex]

[tex]a = 200[/tex]

------------------

[tex]b = 45[/tex]

[tex]a + b + c = 330[/tex]

[tex]245 + c = 330[/tex]

[tex]c = 85[/tex]

The number of people that have the investment plan is:

[tex]C = 85 + 50 = 135[/tex]

135 people have the investment plan

Answer:

You should make 150 quarts of Creamy Vanilla and 50 quarts of Continental Mocha.

Step-by-step explanation:

This problem can be solved by a system of equations.

The largest profit is going to earned when all the eggs and cups of cream in stock are used.

I am going to call x the number of quarts of Creamy Vanilla and y the number of quarts of Continental Mocha.

The problem states that each quart of Creamy Vanilla uses 2 eggs and each quart of Continental Mocha uses 1 egg. There are 350 eggs in stock, so:

[tex]2x + y = 350[/tex]

Each quart of Creamy Vanilla uses 3 cups of cream, as does each quart of Continental Mocha. There are 600 cups of cream in stock.

So:

[tex]3x + 3y = 600[/tex] Simplifying by 3.

[tex]x + y = 200[/tex]

We have the following system

[tex]1)2x + y = 350[/tex]

[tex]2)x + y = 200[/tex]

I am going to multiply 2) by (-1) and then add with 1), so i can eliminate y

[tex]1)2x + y = 350[/tex]

[tex]2)-x - y = -200[/tex]

[tex]2x - x + y - y = 350 - 200[/tex]

[tex]x = 150[/tex]

[tex]x + y = 200[/tex]

[tex]y = 200 - 150 = 50[/tex]

You should make 150 quarts of Creamy Vanilla and 50 quarts of Continental Mocha.

The question regarding the number of quarts of Creamy Vanilla and Continental Mocha for maximum profit can be solved using a method in mathematics called linear programming. It involves setting up inequalities to represent the constraints (number of eggs and cups of cream) and a profit function which we want to maximize. The specific solution will depend on the particular set of constraints and profit per flavor.

This problem can be approached as a linear programming problem. Let x represent the number of quarts of Creamy Vanilla and y represent the number of quarts of Continental Mocha. The constraints from the problem can be represented with the following inequalities, considering the number of eggs and cups of cream:

2x + y ≤ 350 (eggs)

3x + 3y ≤ 600 (cups of cream)

We want to maximize the profit function P which is given by P = 3x + 2y. This can be solved graphically by plotting the constraints on a graph, finding the feasible region and then determining which point in that region provides the maximum value of P.

The solution to this problem can vary depending on the specific set of supplies and the profit of each flavor.

https://brainly.com/question/30763902

#SPJ3

Assume a standard deck of 52 cards with 4 suits of 13 cards each.

a. There are [tex]\dbinom{13}8\dbinom{39}2[/tex] ways of being dealt a hand consisting of 8 hearts and 2 non-hearts, so the probability of being dealt such a hand is

[tex]\dfrac{\dbinom{13}8\dbinom{39}2}{\dbinom{52}{10}}\approx0.0000602823[/tex]

b. This time, the non-hearts specifically belong to the suit of diamonds, for which there are [tex]\dbinom{13}2[/tex] ways of getting drawn, so the probability is

[tex]\dfrac{\dbinom{13}8\dbinom{13}2}{\dbinom{52}{10}}\approx0.0000063455[/tex]

a. The probability P of being dealt exactly 8 hearts is:

[tex]\[ P = \frac{k}{n} = \frac{\binom{13}{8} \times \binom{39}{2}}{\binom{52}{10}} \][/tex]

b. The probability  P'  of being dealt exactly 8 hearts given the first two cards were diamonds is:

[tex]\[ P' = \frac{k'}{n'} = \frac{\binom{8}{6} \times \binom{42}{2}}{\binom{50}{8}} \][/tex]

To solve this problem using combinatorics, we can calculate the probability by considering the total number of possible outcomes and the number of favorable outcomes.

Let's denote:

-  n  as the total number of ways to deal 10 cards from a standard deck (52 cards).

-  k  as the number of ways to deal exactly 8 hearts and 2 non-hearts from the remaining 44 cards in the deck.

a. To calculate the probability a player is dealt exactly 8 hearts:

Total number of ways to choose 8 hearts out of 13 hearts:

[tex]\[ \binom{13}{8} \][/tex]

Total number of ways to choose 2 non-hearts out of 39 non-hearts:

[tex]\[ \binom{39}{2} \][/tex]

Therefore, the number of favorable outcomes is:

[tex]\[ k = \binom{13}{8} \times \binom{39}{2} \][/tex]

The total number of ways to deal 10 cards from a deck of 52 cards is:

[tex]\[ n = \binom{52}{10} \][/tex]

So, the probability P of being dealt exactly 8 hearts is:

[tex]\[ P = \frac{k}{n} = \frac{\binom{13}{8} \times \binom{39}{2}}{\binom{52}{10}} \][/tex]

b. To calculate the probability a player is dealt exactly 8 hearts if the first two cards dealt were diamonds:

If the first two cards are diamonds, then there are 50 cards remaining, out of which 8 are hearts and 42 are non-hearts.

Total number of ways to choose 6 more hearts out of the remaining 8 hearts:

[tex]\[ \binom{8}{6} \][/tex]

Total number of ways to choose 2 non-hearts out of the remaining 42 non-hearts:

[tex]\[ \binom{42}{2} \][/tex]

Therefore, the number of favorable outcomes is:

[tex]\[ k' = \binom{8}{6} \times \binom{42}{2} \][/tex]

The total number of ways to deal 8 cards from the remaining 50 cards is:

[tex]\[ n' = \binom{50}{8} \][/tex]

So, the probability  P'  of being dealt exactly 8 hearts given the first two cards were diamonds is:

[tex]\[ P' = \frac{k'}{n'} = \frac{\binom{8}{6} \times \binom{42}{2}}{\binom{50}{8}} \][/tex]

Answer:

The solution for the initial value problem is:

[tex]y(x) = -5e^{\frac{x^{2}}{2} - 2x} + 10[/tex]

Step-by-step explanation:

We have the following initial value problem:

[tex]\frac{dy}{dx} = (x-2)(y-10)[/tex]

The first step is solving the differential equation. We can do this by the variable separation method. It means that every term with y in on one side of the equality, every term with x on the other side. So:

[tex]\frac{dy}{dx} = (x-2)(y-10)[/tex]

[tex]\frac{dy}{y-10} = (x-2)dx[/tex]

To find y in function of x, we integrate both sides.

[tex]\frac{dy}{y-10} = (x-2)dx[/tex]

[tex]\int {\frac{1}{y-10}} \, dy = \int {(x-2)} \, dx[/tex]

Solving each integral separately

[tex]\int {\frac{1}{y-10}} \, dy[/tex]

This one we solve by substitution

[tex]u = y-10, du = dy[/tex]

[tex]\int {\frac{1}{y-10}} \, dy = \int {\frac{1}{u}} \, du = \ln{u} = \ln{(y-10)}[/tex]

[tex] \int {(x-2)} \, dx = \frac{x^{2}}{2} - 2x + K[/tex]

Now we have that:

[tex]\int {\frac{1}{y-10}} \, dy = \int {(x-2)} \, dx[/tex]

[tex]\ln{(y-10) = \frac{x^{2}}{2} - 2x + K}[/tex]

To solve for y, we apply the exponential to both sides, since the exponential and ln are inverse operations:

[tex]e^{\ln{(y-10)} = e^{\frac{x^{2}}{2} - 2x + K}[/tex]

[tex]y - 10 = Ke^{\frac{x^{2}}{2} - 2x}[/tex]

[tex]y(x) = Ke^{\frac{x^{2}}{2} - 2x} + 10[/tex]

[tex]y(0) = 5[/tex] means that when [tex]x = 0, y(x) = 5[/tex]. So:

[tex]5 = Ke^{\frac{0^{2}}{2} - 2*0} + 10[/tex]

[tex]Ke^{0} = -5[/tex]

[tex]K = -5[/tex]

So, the solution for the initial value problem is:

[tex]y(x) = -5e^{\frac{x^{2}}{2} - 2x} + 10[/tex]

Answer:  80

Step-by-step explanation:

Given : A restaurant serves a luncheon special .

The number of choices for appetizers= 4

The number of choices for salad = 4

The number of choices for desert = 5

Then, the total number of different possible meals are given by :_

[tex]4\times4\times5=80[/tex]

Therefore, the total number of different possible meals are =80

Answer:

Every line in [tex]\mathbb{R}^{3}[/tex] is a function of the form [tex]\gamma (t)={\bf p}+t {\bf v} [/tex], where [tex]{\bf p}[/tex] is point where the line passes and [tex]{\bf v}[/tex] is a nonzero vector which is called the direction vector of the line. Then, if we derive the function [tex]\gamma[/tex] we obtain [tex]\gamma'(t)={\bf v} \neq (0,0,0)[/tex], so [tex]\gamma(t)={\bf p}+t {\bf v}[/tex] is a regular curve.

Step-by-step explanation:

Every line in [tex]\mathbb{R}^{3}[/tex] can be parametrized by

[tex]\gamma (t)={\bf p}+t{\bf v}=(p_{1},p_{2},p_{3})+t(v_{1},v_{2},v_{3})=(p_{1}+tv_{1},p_{2}+tv_{2},p_{3}+tp_{3})[/tex], where [tex]t\in \mathbb{R}[/tex]. To derivate the function [tex]\gamma [/tex] we only need to derive each component. Then we have that

[tex]\gamma'(t)=(\frac{d}{dt}(p_{1}+tv_{1}),\frac{d}{dt}(p_{2}+tv_{2}),\frac{d}{dt}(p_{3}+tv_{3}))=(v_{1},v_{2},v_{3})={\bf v}\neq (0,0,0).[/tex]

Now, remember that a a parametrized curve is said to be regular if [tex]\gamma'\neq 0[/tex] for all [tex]t[/tex].

Area = Length time width.

The area of the shaded area is Length x width of the entire shape, minus the length x width of the unshaded area.

Full area: 27 x 10 = 270 square m.

Unshaded area: 17 x 5 = 85 square m.

Area of shaded region: 270 - 85 = 185 square m.

Smallest number of eggs in the basket = 119

In the question,

We know that the number of eggs in the basket should be a a multiple of 7.

But it can not be a multiple of 2, 3, 4, 5 and 6 because every time we pick up the eggs 2, 3, 4, 5 or 6 at a time we are left with some eggs with us.

Therefore, the number of eggs can not be a multiple of these numbers.

Now,

Let us say the number of eggs in the basket be 7x.

So,

Let us take the LCM of 2, 3, 4, 5 and 6.

So,

LCM = 60

Now, the number would be greater than 60 and the multiple of 7.

So, checking on all the multiples of 7 above 60 and checking the condition that, the remainder left on dividing by,

2 is 1. (2x + 1)

by 3 is 2. (3x + 2)

by 4 is 3. (4x + 3)

by 5 is 4. (5x + 4)

by 6 is 5. (6x + 5)

So,

On Checking the multiples of 60 which are divisible by 7 are ,

60 + 1 , 120 + 1, 60 - 1, 120 - 1.

So,

For 61 it is not satisfying all the conditions.

121 is also not satisfying all the conditions.

59 is also not satisfying all the conditions.

But,

The number 119 on checking its divisibility by 2. Leaves remainder 1 as, (2(59) + 1).

Divisibility by 3 leaves remainder 2 as, {3(39) + 2}.

Divisibility by 4 leaves remainder 3 as, {4(29) + 3}.

Divisibility by 5 leaves remainder 4 as, {5(23) + 4}.

Divisibility by 6 leaves remainder 5 as, {6(19) + 5}.

Divisibility by 7 leaves remainder 0 as, {7(17) + 0}.

Therefore, the minimum number of eggs in the basket are 119.

Answer: 119 is the smallest number of eggs in the  basket

Step-by-step explanation:

let k be the number

so k is less than a multiple of 2 by 1 so we have k = 2A - 1 for a positive integer  A

so k is 2 more and thus less with 1, for a multiple of 3, k = 3B - 1, for some positive integer B

so k is 3 more and thus less with 1, with a multiple of 4, so k = 4C -1 for some positive integer

so k is 4 more and thus less with 1, with a multiple of 5, so k = 5D -1 for some positive integer

so k is 5 more and thus less with 1, with a multiple of 6, so k = 6E -1 for some positive integer

thus k is a multiple of 7, so k = 7F for some positive integer F

THUS K = 2A - 1 = 3B - 1 = 4C -1 = 5D -1 = 6E -1 = 7F

ADD 1 to k above gives k + 1 = 2A = 3B = 4C = 5D = 6E = 7F + 1

of k + 1 = 7F + 1 has to be a common multiple of 2, 3, 4, 5, 6. The least common multiple  of 2, 3, 4, 5, 6 is 60

thus k + 1 is a multiple of 60 so k = 60 -1 = 59

again  find the least multiple of 60 that is 1 greater  than a multiple of  60

the first multiple of 60 is 60 and 60 - 1 = 59, but 59 is not a multiple of 7

the second multiple of 60 is 120 and 120 -1 = 119

and 7 is a multiple of 119 because 7 * 17 = 119

Answer:

Distributive property

Subtraction property of equality

Division property of equality

Step-by-step explanation:

Given

-6(x-4)=42

-6*x+-4*-6=42-------------------distributive property

-6x+24-24=42-24-----------------subtraction property of equality

-6x=18

-6x/-6=18/-6--------------------------division property of equality

x= -3

Answer: Fraction of all eligible voters voted for Shelly is [tex]\dfrac{3}{20}[/tex] .

Morgan received the most votes .

Step-by-step explanation:

Given : The fraction of all eligible voters cast votes =[tex]\dfrac{1}{2}[/tex]

The fraction of votes received by Shelly = [tex]\dfrac{3}{10}[/tex]        (1)

Now, the fraction of all eligible voters voted for Shelly is given by :_

[tex]\dfrac{3}{10}\times\dfrac{1}{2}=\dfrac{3}{20}[/tex]

Thus, the fraction of all eligible voters voted for Shelly is [tex]\dfrac{3}{20}[/tex] .

The fraction of vote received by Morgan= [tex]\dfrac{5}{8}[/tex]          (2)

To compare the fractions given in (1) and (2), we need to find least common multiple of 10 and 8 .

LCM (10, 8)=40

Now, make denominator 40 (to make equivalent fraction)  in (1), (2) we get

Fraction of all eligible voters voted for Shelly = [tex]\dfrac{3\times4}{10\times4}=\dfrac{12}{40}[/tex]

Fraction of all eligible voters voted for Morgan =[tex]\dfrac{5\times5}{8\times5}=\dfrac{25}{40}[/tex]

Since, [tex]\dfrac{25}{40}>\dfrac{12}{40}[/tex]  [By comparing numerators]

Therefore, Morgan received the most votes .

Answer: 0.002718

Step-by-step explanation:

Given : The population mean annual salary for environmental compliance specialists is about ​$62,000.

i.e. [tex]\mu=62000[/tex]  

Sample size : n= 32

[tex]\sigma=6200[/tex]

Let x be the random variable that represents the annual salary for environmental compliance specialists.

Using formula [tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex], the z-value corresponds to x= 59000 will be :

[tex]z=\dfrac{59000-62000}{\dfrac{6200}{\sqrt{32}}}\approx\dfrac{-3000}{\dfrac{6200}{5.6568}}=-2.73716129032\approx-2.78[/tex]

Now, by using the standard normal z-table , the probability that the mean salary of the sample is less than ​$59,000 :-

[tex]P(z<-2.78)=0.002718[/tex]

Hence, the probability that the mean salary of the sample is less than ​$59,000= 0.002718

Answer:

Inverse Laplace of [tex]\frac{1}{(S+4)^2}[/tex] will be [tex]te^{-4t}[/tex]

Step-by-step explanation:

We have to find the inverse Laplace transform of [tex]H(S)=\frac{1}{(S+4)^2}[/tex]

We know that of [tex]\frac{1}{s+4}[/tex] is [tex]e^{-4t}[/tex]

As in H(s) there is square of [tex]s+4[/tex]

So i inverse Laplace there will be multiplication of t

So the inverse Laplace of [tex]\frac{1}{(s+4)^2}[/tex]  will be [tex]te^{-4t}[/tex]

[tex]L^{-1}\frac{1}{(S+4)^2}=te^{-4t}[/tex]

Answer:

The linear equation relating price to demand is [tex]p=-0.05x+360.2[/tex]

Step-by-step explanation:

A Linear Demand Function expresses demand p (the number of items demanded) as a function of the unit price x (the price per item).

From the information given we know two facts:

Let x be the price and p the number of cards sold.

The fact 1. is the slope of the function because is the change in demand per unit change in price.

We can use fact 2. to find the value of b in the equation

[tex]p=-0.05x+b\\b=p+0.05x[/tex]

[tex]b=360+0.05\cdot 4\\b=360+0.2\\b=360.2[/tex]

The linear equation relating price to demand is [tex]p=-0.05x+360.2[/tex]

Step-by-step explanation:

As the hint says, for any function [tex]f:\Omega\to\mathcal{P}(\Omega)[/tex], we can think of the set [tex] X=\{ \omega\in\Omega : \omega \notin f(\omega)\}[/tex] (which is the set of all those elements of [tex]\Omega[/tex] which don't belong to their image). So [tex]X[/tex] is made of elements of [tex]\Omega[/tex], and so it belongs to [tex]\mathcal{P}(\Omega)[/tex].

Now, this set [tex]X[/tex] is NOT the image of any element in [tex] \Omega[/tex], since if there was some [tex]a\in\Omega[/tex] such that [tex]f(a)=X[/tex], then the following would happen:

If [tex]a\in X=f(a)[/tex], then by definition of the set [tex]X[/tex], [tex]a\notin f(a)[/tex], so we're getting that [tex]a\in f(a)[/tex] and also [tex] a\notin f(a)[/tex], which is a contradiction.

On the other hand, if [tex]a\notin f(a)[/tex], then by definition of the set [tex]X[/tex], we would get that [tex]a\in X=f(a)[/tex], so we're getting that [tex]a\in f(a)[/tex] and also [tex] a\notin f(a)[/tex], which is a contradiction again.

So in any case, the assumption that this set [tex]X[/tex] is the image of some element in [tex]\Omega[/tex] leads us to a contradiction, therefore this set [tex]X[/tex] is NOT the image of any element in [tex]\Omega[/tex], and so there cannot be a bijection from [tex]\Omega[/tex] to [tex]\mathcal{P}(\Omega)[/tex], and so the two sets cannot have the same cardinality.

Answer:

The possible combinations are (4 & 6) ,(8 &3), (2 &12), (24 &1)

Step-by-step explanation:

The area of a flower bed is 24 square feet.

Now, the factors of 24 are 2 x 2 x 2 x 3.

Hence, if the other sides were whole number dimensions, then the possible combinations will be (4 & 6) ,(8 &3), (2 &12), (24 &1)

Answer:

0.2449 or 24.49%

Step-by-step explanation:

Calculate the IRR if the power company gets a fixed feed-in tariff of $0.25/kWh.

IRR means Internal Rate of Return is given by

[tex]IRR=(\frac{FV}{PV})^{\frac{1}{n}}-1[/tex]

Where,  FV = Final value ($20)

             PV = Present value ($0.25)

               N = 20 years

Now put the values

[tex]IRR=(\frac{20}{0.25})^{\frac{1}{20}}-1[/tex]

= [tex]IRR=(80)^{\frac{1}{20}}-1[/tex]

= 1.24495742 - 1

= 0.24495742

Converting in percentage :

0.24495742 × 100 = 24.49%

IRR = 0.2449 or 24.49%

Answer:

IRR = 0.24495

Step-by-step explanation:

Given data:

Tariff =$0.25

So, present value = $0.25

N = 20 year

salvage value after 20 year is $20 M

final value is $20 M

IRR means internal rate of return and it is given as

[tex]IRR =[\frac{FV}{PV}]^{1/n} -1[/tex]

Where FV is final value and PV is present value

[tex]IRR = [\frac{20}{0.25}]^{1/20} -1[/tex]

IRR = 0.24495

Answer:

No

.3 = .30

.32 > .30

Step-by-step explanation:

Answer:

The solutions are: [tex]x=4,\:x=-4,\:x=3i,\:x=-3i[/tex]

Step-by-step explanation:

Consider the provided equation.

[tex]x^4-7x^2-144=0[/tex]

Substitute [tex]u=x^2\mathrm{\:and\:}u^2=x^4[/tex]

[tex]u^2-7u-144=0[/tex]

[tex]u^2-16u+9u-144=0[/tex]

[tex](u-16)(u+9)=0[/tex]

[tex]u=16,\:u=-9[/tex]

Substitute back [tex]\:u=x^2[/tex] and solve for x.

[tex]x^2=16\\x=\sqrt{16}\\ \quad x=4,\:x=-4[/tex]

Or

[tex]x^2=-9\\x=\sqrt{-9}\\ \quad x=3i,\:x=-3i[/tex]

Hence, the solutions are: [tex]x=4,\:x=-4,\:x=3i,\:x=-3i[/tex]

Check:

Substitute x=4 in provided equation.

[tex]4^4-7(4)^2-144=0[/tex]

[tex]256-112-144=0[/tex]

[tex]0=0[/tex]

Which is true.

Substitute x=-4 in provided equation.

[tex](-4)^4-7(-4)^2-144=0[/tex]

[tex]256-112-144=0[/tex]

[tex]0=0[/tex]

Which is true.

Substitute x=3i in provided equation.

[tex](3i)^4-7(3i)^2-144=0[/tex]

[tex]81+63-144=0[/tex]

[tex]0=0[/tex]

Which is true.

Substitute x=-3i in provided equation.

[tex](-3i)^4-7(-3i)^2-144=0[/tex]

[tex]81+63-144=0[/tex]

[tex]0=0[/tex]

Which is true.

Answer:

It is sufficient to prove that  [tex] p_1\implies p_2, p_2\implies p_3, p_3\implies p_1[/tex]

Step-by-step explanation:

The propositions [tex] p_1,p_2,p_3[/tex] being equivalent means they should always have the same truth value. If one of them is true, then all of them must be true. And if one of them is false, then all of them must be false.

Suppose we've proven that [tex] p_1\implies p_2, p_2\implies p_3, p_3\implies p_1[/tex] (call these first, second and third implications).

If [tex]p_1[/tex] was true, then by the first implication that we proved, it would follow that [tex]p_2[/tex] is also true. And then by the second implication that we prove it would follow then that [tex]p_3 [/tex] is also true. Therefore the three of them would be true. Notice the reasoning would have been the same if we had started assuming that the one that was true was either [tex]p_2~or~p_3[/tex]. So one of them being true makes all of them be true.

On the other hand, if [tex]p_1[/tex] was false, then by the third implication that we proved, it would follow that [tex]p_3[/tex] has to be false (otherwise [tex]p_1[/tex] would have to be true, which would be a contradiction). And then, since [tex]p_3[/tex] is false, by the second implication that we proved it would follow that [tex] p_2[/tex] is false (otherwise [tex] p_3[/tex] would have to be true, which would be a contradiction). Therefore the three of them would be false. Notice the reasoning would have been the same if we had started assuming that the one that was false was either [tex]p_2~or~p_3[/tex]. So one of them being false makes all of them be false.

So, the three propositions always have the same truth value, and so they're all equivalent.

Answer:

88

Step-by-step explanation:

We are given that in arithmetic sequence , the nth term [tex]a_n[/tex] is given by the formula

[tex]A_n=a_1+(n-1)d[/tex]

Where [tex]a_1=first term[/tex]

d=Common difference

In an geometric sequence, the nth term is given by

[tex]a_n=a_1r^{n-1}[/tex]

Where r= Common ratio

1,4,7,10,..

We have to find 30th term.

[tex]a_1=1,a_2=4,a_3=7,a_4=10[/tex]

[tex]d=a_2-a_1=4-1=3[/tex]

[tex]d=a_3-a_2=7-4=3[/tex]

[tex]d=a_4-a_3=10-7=3[/tex]

[tex]r_1=\frac{a_2}{a_1}=\frac{4}{1}=4[/tex]

[tex]r_2=\frac{a_3}{a_2}=\frac{7}{4}[/tex]

[tex]r_1\neq r_2[/tex]

Therefore, given sequence is an arithmetic sequence because the difference between consecutive terms is constant.

Substitute n=30 , d=3 a=1 in the given formula of arithmetic sequence

Then, we get

[tex]a_{30}=1+(30-1)(3)=1+29(3)=1+87=88[/tex]

Hence, the 30th term of sequence is 88.

Answer:

y = 2

Step-by-step explanation:

Answer:

y=2

Step-by-step explanation:

Final answer:

The correct negation of the statement is option C: 'This triangle has two 45 degree angles and it is not a right triangle,' which follows the 'P and not Q' format for negating 'If P, then Q' statements.

Explanation:

The negation of the statement 'If this triangle has two 45 degree angles then it is a right triangle' is 'This triangle has two 45 degree angles and it is not a right triangle.' To negate a conditional statement like the one given, you would state that while the condition holds true, the conclusion does not. This is represented in option C. When negating an 'if-then' statement, the format usually follows 'If P, then Q' to 'P and not Q.' Thus, the correct answer is C, 'this triangle has two 45 degree angles and it is not a right triangle.'

Answer:

[tex]10.897[/tex]

Step-by-step explanation:

An baseball player’s batting average decreases from 0.312 to 0.278 .

Let [tex]x_0[/tex] be the initial  baseball player’s batting average and [tex]x_1[/tex] be the final baseball player’s batting average .

Initial value [tex]\left ( x_0 \right )[/tex] = 0.312

Final value [tex]\left ( x_1 \right )[/tex] = 0.278

So, change in value =Final value - Initial value =  [tex]x_1-x_0[/tex] = [tex]0.278-0.312=-0.034[/tex]

Therefore , decrease in value = 0.034

We know that  percent decreased = ( decrease in value × 100 ) ÷ Initial value

i.e. , [tex]\frac{0.034}{0.312}\times 100=\frac{3400}{312}=10.897[/tex]

Or we can say percentage change in  baseball player’s batting average = [tex]-10.897 \%[/tex]

Answer:

[tex]x=\frac{-31\pm \sqrt{1000.2}}{4}[/tex]

Step-by-step explanation:

Given quadratic equation,

[tex]2x^2+31x-4.9=0[/tex]

Since, by the quadratic formula,

The solution of a quadratic equation [tex]ax^2+bx+c=0[/tex] is,

[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]

Here, a = 2, b = 31, c = -4.9,

Thus, by the quadratic formula,

[tex]x=\frac{-31\pm \sqrt{31^2-4\times 2\times -4.9}}{2\times 2}[/tex]

[tex]=\frac{-31\pm \sqrt{961+39.2}}{4}[/tex]

[tex]=\frac{-31\pm \sqrt{1000.2}}{4}[/tex]

[tex]\implies x = \frac{-31+\sqrt{1000.2}}{4}\text{ or }x=\frac{-31- \sqrt{1000.2}}{4}[/tex]

Answer:  Option 'c' is correct.

Step-by-step explanation:

Since we have given that

Number of ounces steaks = 4.5 oz

Number of guests = 180

Percentage of waste = 25%

Number of ounces of fresh steak is given by

[tex]4.5\times 180\\\\=810\ oz[/tex]

Let the number of ounces of raw steak be x.

According to question,

[tex]\dfrac{100-25}{100}\times x=810\\\\\dfrac{75}{100}\times x=810\\\\0.75\times x=810\\\\x=\dfrac{810}{0.75}\\\\x=1080\ oz[/tex]

As we need in pounds, and we know that

1 pound = 16 ounces

So, Number of pounds of raw steak that Chef could order is given by

[tex]\dfrac{1080}{16}=67.5\ lbs[/tex]

Hence, Option 'c' is correct.

To determine how many pounds of raw steak Chef should order for 180 guests with 25% waste, calculate the weight per guest and multiply by the number of guests. Convert the weight to pounds by dividing by 16. The answer is approximately B. 63.3 lbs.

To determine how many pounds of raw steak Chef should order, we need to calculate the total weight of steak required for 180 guests, taking into account the 25% waste.

First, we calculate the weight of steak for one guest by multiplying 4.5 oz by 1.25 (to account for the waste), which equals 5.625 oz.

Then, we multiply the weight per guest by the number of guests to get the total weight of steak required: 5.625 oz x 180 = 1012.5 oz.

Finally, we convert the ounces to pounds by dividing by 16, since there are 16 ounces in a pound.

Thus, Chef should order 1012.5 oz ÷ 16 = 63.28125 lbs, which rounds to approximately 63.3 lbs (option B).

Hi here´s a way to solve it

Let m and n be odd integers. Then,  we can express m as 2r + 1 and n as 2s + 1, where r and s are integers.

This means that any odd number can be written as the sum of some even integer and one.

Substituting, we have that m + n = (2r + 1) + 2s + 1 = 2r + 2s + 2.

As we defined r and s as integers, 2r + 2s + 2 is also an integer.

Now It is clear that 2r + 2s + 2 is an integer divisible by 2 becasue we have 2 in each of the integers.

Therefore,  2r + 2s + 2 = m + n is even.

So, the sum of two odd integers is even.

Answer:

There are 52 dimes and 18 nickles

Step-by-step explanation:

Lets call x= number of dimes and y= number of nickles

then we have the first equation  

(1)   x + y = 70

As a nickel is worth 0.05 US$ and a dime is worth 0.10 US$, we have the second equation

(2)   0.10x + 0.05y = 6.10

We then have a linear system of 2 equations and 2 unknowns  

(1) x +y = 70

(2) 0.10x + 0.05y = 6.10

In order to solve the system by the elimination method, we have to multiply on of the equations by a suitable number to eliminate one unknown when adding the two equations.  

There are several ways of doing this. We could, for example, multiply (1) by -0.05  and then add it to (2)

(1) -0.05x – 0.05y = (-0.05)70

(2) 0.10x + 0.05y = 6.10

That is to say,

(1) -0.05x – 0.05y = -3.5

(2) 0.10x + 0.05y = 6.10

Adding (1) and (2) we get

-0.05x = -2.6 => x = (-2.6)/(-0.05) = 52 => x = 52

So we have 52 dimes.

Substituting this value in equation 1, we obtain

  y = 70 - x = 70 – 52 = 18

Then we have 18 nickels

The  difference in the cost for a one-night stay with and without breakfast is:

                             $ 23

Total cost to stay one night at sleepyhead Motel without breakfast is: $119.

and cost to stay one night at sleepyhead Motel with breakfast is: $142.

Hence, the difference in the cost for a one-night stay with and without breakfast is calculated by:

               $ (142-119)

                 =   $ 23

Hence, the answer is: $ 23