A shop sells apple juice in cartons of two sizes. A 500 ml carton costs $1.30.
a) Work out the cost of 250 ml of this apple juice. A 750 ml carton costs $1.86.
b) Work out the cost of 250 ml of this apple juice.
c) Which carton is better value for money?

Answer:

Part A)

[tex]\displaystyle \frac{dy}{dx}=-\frac{2xy+y^2}{x^2+2xy}[/tex]

Part B)

[tex]\displaystyle y=-\frac{5}{8}x+\frac{9}{4}[/tex]

Step-by-step explanation:

We have the equation:

[tex]\displaystyle x^2y+y^2x=6[/tex]

Part A)

We want to find the derivative of our function, dy/dx.

So, we will take the derivative of both sides with respect to x:

[tex]\displaystyle \frac{d}{dx}\Big[x^2y+y^2x\Big]=\frac{d}{dx}\big[6\big][/tex]

The derivative of a constant is 0. We can expand the left:

[tex]\displaystyle \frac{d}{dx}\Big[x^2y\Big]+\frac{d}{dx}\Big[y^2x\Big]=0[/tex]

Differentiate using the product rule:

[tex]\displaystyle \Big(\frac{d}{dx}\big[x^2\big]y+x^2\frac{d}{dx}\big[y\big]\Big)+\Big(\frac{d}{dx}\big[y^2\big]x+y^2\frac{d}{dx}\big[x\big]\Big)=0[/tex]

Implicitly differentiate:

[tex]\displaystyle (2xy+x^2\frac{dy}{dx})+(2y\frac{dy}{dx}x+y^2)=0[/tex]

Rearrange:

[tex]\displaystyle \Big(x^2\frac{dy}{dx}+2xy\frac{dy}{dx}\Big)+(2xy+y^2)=0[/tex]

Isolate the dy/dx:

[tex]\displaystyle \frac{dy}{dx}(x^2+2xy)=-(2xy+y^2)[/tex]

Hence, our derivative is:

[tex]\displaystyle \frac{dy}{dx}=-\frac{2xy+y^2}{x^2+2xy}[/tex]

Part B)

We want to find the equation of the tangent line at (2, 1).

So, let's find the slope of the tangent line using the derivative. Substitute:

[tex]\displaystyle \frac{dy}{dx}_{(2,1)}=-\frac{2(2)(1)+(1)^2}{(2)^2+2(2)(1)}[/tex]

Evaluate:

[tex]\displaystyle \frac{dy}{dx}_{(2,1)}=-\frac{4+1}{4+4}=-\frac{5}{8}[/tex]

Then by the point-slope form:

[tex]y-y_1=m(x-x_1)[/tex]

Yields:

[tex]\displaystyle y-1=-\frac{5}{8}(x-2)[/tex]

Distribute:

[tex]\displaystyle y-1=-\frac{5}{8}x+\frac{5}{4}[/tex]

Hence, our equation is:

[tex]\displaystyle y=-\frac{5}{8}x+\frac{9}{4}[/tex]

Answer:

69 feet

Step-by-step explanation:

Now go ace that test!

Answer: 69

Step-by-step explanation:

Answer: one with change mane I’m pretty sure? Really sorry if that’s not right

Step-by-step explanation:

Answer:

I’m pretty sure it’s 7. Just divide the 35 by 5 bc 5 times 3 is 15 and so it would be 7

Hope this helps

Answer:

c is the correct answer

Answer:

Step-by-step explanation:

Here are the missing values;

Mean μ = 15.5 minutes

Standard deviation = 1.7 minutes

A Random sample of 90 completion

The sample mean = 15.4 minutes

Level of significance = 0.1

Then the following analysis can be made on the above study.

Firstly, the null hypothesis is [tex]\mathbf{H_o : \mu = 15.5}[/tex]

the alternative hypothesis is [tex]\mathbf{H_a: \mu < 15.5}[/tex]  

Since, the value is less than, then this is a one-tailed test.

The Z test statistics can be computed as:

[tex]Z = \dfrac{ \overline x - \mu }{\dfrac{\sigma}{\sqrt{n}} } \ \ \sim \ \ N(0.1)[/tex]

[tex]Z = \dfrac{ 15.4-15.5 }{\dfrac{1.7}{\sqrt{90}} }[/tex]

[tex]Z = \dfrac{ -0.1 }{\dfrac{1.7}{ 9.4868} }[/tex]

Z = −0.560

The critical value of Z at 0.1  level  of significance is:

[tex]Z_{0.1} = -1.28[/tex]

Decision Rule: We fail to reject the null hypothesis sInce -0.560 > -1.28

Conclusion: NO, there is no evidence to support the claim that the mean completion time has decreased. We conclude that the mean completion time remains at 15.5 minutes.

Step-by-step explanation:

the slope if the equation will be -8

Answer:

Much larger than 1 for both of them

Step-by-step explanation:

Answer:

8,20,24

Step-by-step explanation:

these numbers they make 52

Answer:

Answer: The minimum number of full hours did Carson need to work at his father's business is 8 hours and 15 minutes. Happy to help!

Step-by-step explanation:

Answer:

360 cookies

Step-by-step explanation:

60 times 6= 360

Answer:

8 :)

Step-by-step explanation:

Answer:

2.8 + 27.44k

Step-by-step explanation:

Algebraic Simplification

The rules of algebra can be applied in some situations to make expressions simplified or expanded, according to the specific needs of each problem.

We are given the expression:

(1 + 9.8k)(2.8)

It's a sum and a product. We cannot add the independent term with the variable term, so we can only multiply the factor and the contents of the parentheses as follows:

(2.8)1 + (2.8)9.8k

Operating:

2.8 + 27.44k

Answer:

33  7/9 or 304/9

Step-by-step

Answer:

The person above is right give them brainliest please!

Step-by-step explanation:

Answer:

a 5 digit number

Step-by-step explanation:

Pick the largest three digit number (999) and the largest two digit number (99) and multiply them to get 999*99 = 98901

The result is a five digit number.

Answer: 5 digit number

Answer:

no, it's not.

Step-by-step explanation:

300- 284 = 16

284 - 236 = 48

236 - 156 = 80

156 - 44 = 112

for every one the top line goes up, the bottom line changes too, but it's not at a constant rate. it's not constant, therefore it cannot be linear.

the rate of change formula (or slope formula, they're the same thing) is :

y2- y1/ x2 -x1 and to solve it you plug in the points. x is the same a t and y is h

284-300/ 1-0

-16 /1

-16 (this is the rate of change between the first two points)

236- 284/ 2-1

-48/ 1

-48 (rate of change between the second and third points)

since the rate of change isn't constant, it's not possible for it to be a linear relationship.

Answer:

I  D  K

Step-by-step explanation:

my brains not working right now

Answer:

thats not a question

Step-by-step explanation:

===========================================

Work Shown:

RS*QS = (TS)^2

RS*(QR+RS) = (TS)^2

27*(5x-12+27) = (45)^2

27*(5x+15) = 2025

135x+405 = 2025

135x = 2025-405

135x = 1620

x = 1620/135

x = 12

Use this to find the length of QR, which is the same as RQ

QR = 5x-12

QR = 5*12-12

QR = 60-12

QR = 48

9514 1404 393

Answer:

  RQ = 48

Step-by-step explanation:

The product of secant segment lengths to the points of intersection with the circle is the same for both secants. The tangent is a degenerate case where the two points of intersection are the same.

  ST·ST = SR·SQ

  SQ = ST^2/SR = 45^2/27 = 75

  RQ = SQ -SR = 75 -27

  RQ = 48

Answer:

12, 24, 36, 48, 60, 72

Step-by-step explanation:

Answer:

ok its c daddy hope it helps dad

Step-by-step explanation:

The probability is (0.10 * 0.50 + 0.01 * 0.20 + 0.89 * 0.01) * 100% = 6.09%.

Answer:

[tex]\sin(C)=\pm\sqrt{1-\cos^2(C)}[/tex]

Step-by-step explanation:

Given the Pythagorean Identity:

[tex]\sin^2(C)+\cos^2(C)=1\\[/tex]

We want to solve for sin(C).

First, we will subtract cos²(C) from both sides:

[tex]\sin^2(C)=1-\cos^2(C)[/tex]

Next, we will take the square root of both sides. Since we are taking an even-root, we will need to add plus/minus. Hence:

[tex]\sin(C)=\pm\sqrt{1-\cos^2(C)}[/tex]

Answer:

21.3 unit

Step-by-step explanation:

[tex]side CB=\sqrt{(7^{2})+(2^{2})} =\sqrt{53}\\side CA=\sqrt{(5)^{2}+(2^{2})} =\sqrt{29} \\side AB=\sqrt{(5^{2} )+(7^{2} )} =\sqrt{74}[/tex]

[tex]Perimeter=\sqrt{53} +\sqrt{29} + \sqrt{74} =21.2676[/tex] unit

Answer: 280

Step-by-step explanation:

Remember the rounding rule -

If it is 5 or more, you round it up, if it is 4 or less, you round it down.

283 rounded to the nearest ten is 280.

Answer:

750 miles

Step-by-step explanation:

Step one:

Given data

we are told that the hummer gets to 300miles on 20 gallons of gas

Required

The distance that the Hummer can go on 50 gallons of gas

Step two:

if 20 gallons gets to 300miles

the 50 gallons will get to x miles

cross multiply we have

[tex]x= 50*300/20\\\\x=15000/20\\\\x=750 miles[/tex]

Answer:

y = 2x - 5

Step-by-step explanation:

[tex]y = 2(x− 3) + 1 \\ \\ y = 2x - 6 + 1 \\ \\ y = 2x - 5 \\ [/tex]

Answer:

6.3 trees.

Step-by-step explanation:

We have been given that Meryl needs to cut down 10.5 trees for every 5 cabins she builds.

Let us find the number of trees Meryl needs to cut down to build 1 cabin by dividing 10.5 by 5.

Now let us multiply 2.1 by 3 to find the number of trees Meryl needs to cut down to build 3 cabins.

Therefore, Meryl needs to cut down 6.3 trees, if she builds 3 cabins.

Answer:

k = 5

Step-by-step explanation:

First, add 3 to both sides;

[tex]7k = 35[/tex]

Divide by 7:

[tex]\frac{7}{7}k = \frac{35}{7}\\k = 5[/tex]

Now, let's check our answer:

[tex]7(5) - 3 = 32\\35 - 3 = 32\\32 = 32[/tex]

I hope this helps!!

If you have any questions feel free to ask!

- Kay :)

Answer:

Null Hypothesis : P = 0.73

Alternative Hypothesis : P≠ 0.73

Step-by-step explanation:

Explanation:-

According to a recent survey, 73% of college students did not use student loans to pay for college

Let p be the proportion of college students who did not use student loans to pay for college

Population proportion  P = 73% or 0.73

Null Hypothesis :  P = 0.73

Alternative Hypothesis : P≠ 0.73

Answer: potential to kinetic

Step-by-step explanation:

Answer:

Kinetic Energy is defined as energy that a body possesses by virtue of being in motion.

Kinetic Energy is defined as energy that a body possesses by virtue of being in motion.As you walk down stairs, as water boils, and as the planets orbit the sun, kinetic energy is being utilized.GRAVITATIONALApple falls from tree and rolls behind the trunk.

GRAVITATIONALApple falls from tree and rolls behind the trunk.An object’s relative location to a gravitational field is the gravitational potential an object has to do work. Earth’s gravity will pull a ripe apple from the branch and convert it into gravitational kinetic energy.NUCLEARElectrons circle a nucleus at the atomic level.

NUCLEARElectrons circle a nucleus at the atomic level.The mass of an atom’s nucleus is almost entirely comprised of its protons and neutrons. The force binding them together is considered potential energy because it causes them to weigh less than the sum of their parts.ELASTICSpring animation expands and compresses.

ELASTICSpring animation expands and compresses.Elastic potential energy is present when the object, such as a spring, is in a deformed state but can return to a static state. This means the spring has elastic potential energy when it is either stretched or compressed.CHEMICALBattery animation motions towards a full charge.

CHEMICALBattery animation motions towards a full charge.Every animate and inanimate object is the result of chemical reactions. Chemicals in a battery have the potential to be converted into motion, therefore they have chemical potential energy.Potential Energy is defined as the energy stored by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors.

Potential Energy is defined as the energy stored by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors.When you (or a rock) are standing at the top of a hill, you possess more potential energy than when standing at the bottom. This is because your position relative to the Earth’s center of gravity can exert potential energy upon you to bring you closer to that center.Energy can neither be created nor destroyed, this is the conservation of energy law. However, energy can be altered from one form to another.

Energy can neither be created nor destroyed, this is the conservation of energy law. However, energy can be altered from one form to another.All forms of energy are either potential or kinetic energy. Potential refers to stored energy while kinetic is energy in motion.

Energy can neither be created nor destroyed, this is the conservation of energy law. However, energy can be altered from one form to another.All forms of energy are either potential or kinetic energy. Potential refers to stored energy while kinetic is energy in motion.All energy, whether potential or kinetic, is measured in Joules (J).

Answer:

-4

Step-by-step explanation:

if you switch it you get 12-16 and you have to get a negative so 16-12 is 4 so switch it to a nrgative to get -4