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3 years ago

Explain the difference between a rate, a ratio, and a proportion?

Mathematics

1 answer:

kaheart [24]3 years ago

3 0

Answer:

A proportion is an equality of two ratios.

Example : $\frac{1}{3}$ = $\frac{x}{9}$

We write proportions to help us find equivalent ratios and solve for unknown quantities.

A rate is the quotient of a ratio where the quantities have different units.

Example : $\frac{distance}{time}$

A ratio is a comparison of two quantities.

Example : 1 : 3 or $\frac{1}{3}$

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Find the absolute maximum and absolute minimum of the function f(x,y)=2x3+y4f(x,y)=2x3+y4 on the region {(x,y)|x2+y2≤64}{(x,y)|x

ddd [48]

Check where the first-order partial derivatives vanish to find any critical points within the given region:

$f(x,y)=2x^3+y^4\implies\begin{cases}f_x=6x^2=0\\f_y=4y^3=0\end{cases}\implies(x,y)=(0,0)$

The Hessian for this function is

$\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}12x&0\\0&12y^2\end{bmatrix}$

with $\det\mathbf H(0,0)=0$ , so unfortunately the second partial derivative test fails. However, if we take $x=0$ we see that $f(x,y)>0$ for different values of $y$ ; if we take $y=0$ we see $f(x,y)$ takes on both positive and negative values. This indicates (0, 0) is neither the site of an extremum nor a saddle point.

Now check for points along the boundary. We can parameterize the boundary by

$(x,y)=(8\cos t,8\sin t)$

with $0\le t\le2\pi$ . This turns $f(x,y)$ into a univariate function $F(t)$ :

$F(t)=f(8\cos t,8\sin t)=2^{10}\cos^3t+2^{12}\sin^4t$

$\implies F'(t)=3\cdot2^{10}\cos^2t(-\sin t)+2^{14}\sin^3t\cos t=2^{10}\sin t\cos t(16\sin^2t-3\cos t)$

$F'(t)=0\implies\begin{matrix}\sin t=0\implies t=0,\,t=\pi\\\\\cos t=0\implies t=\dfrac\pi2,\,t=\dfrac{3\pi}2\\\\16\sin^2t-3\cos t=0\implies t=2\tan^{-1}\sqrt{\dfrac{\sqrt{1033}-32}3},\,t=2\pi-2\tan^{-1}\sqrt{\dfrac{\sqrt{1033}-32}3}\end{matrix}$

At these critical points, we get

$F(0)=1024$

$F\left(2\tan^{-1}\sqrt{\dfrac{\sqrt{1033}-32}3}\right)\approx893$

$F\left(\dfrac\pi2\right)=4096$

$F(\pi)=-1024$

$F\left(\dfrac{3\pi}2\right)=4096$

$F\left(2\pi-2\tan^{-1}\sqrt{\dfrac{\sqrt{1033}-32}3}\right)\approx893$

We only care about 3 of these results.

$t=\dfrac\pi2\implies(x,y)=(0,8)$

$t=\pi\implies(x,y)=(-8,0)$

$t=\dfrac{3\pi}2\implies(x,y)=(0,-8)$

So to recap, we found that $f(x,y)$ attains

a maximum value of 4096 at the points (0, 8) and (0, -8), and
a minimum value of -1024 at the point (-8, 0).

6 0

4 years ago

How many minutes are in 9 weeks

m_a_m_a [10]

10,080 because you can multiply it out....

4 0

3 years ago

Read 2 more answers

During a sale, a shop offered a discount of 10% on the marked prices of all the items. What

Snowcat [4.5K]

Answer:

₹2835

Step-by-step explanation:

Working out discount of 10% of jeans at ₹1450:

10% of 1450 = 1450/10 = ₹145

the price would now be ₹1450 - ₹145 = ₹1305

Working out discount of 10% of a shirt at ₹850:

10% of 850 = 850/10 = ₹85

the price would now be ₹850 - ₹85 = ₹765

so the price of two shirts = ₹765 × 2 = ₹1530

the total the customer would have to pay:

₹1305 + ₹1530 = ₹2835

8 0

3 years ago

Read 2 more answers

*80 POINTS!!!! PLEASE HELP ME I HAVE BEEN WAITING FOR MORE THAN AN HOUR!!!!!!!!!! MARKING AS BRAINLIEST!! PLEASE SHOW WORK AND E

ruslelena [56]

Answer:

D. tan(-π/6)

Step-by-step explanation:

The tangent function is periodic with a period of π, so ...

tan(5π/6) = tan(5π/6 ± kπ) . . . for any integer k

For k = -1, we have ...

tan(5π/6) = tan(5π/6 -π) = tan(-π/6)

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3 years ago

What is the interquartile range of the numbers 9, 28, 16, 2, 33, 6, 10

Mumz [18]

Answer:

IQR = 22

Step-by-step explanation:

IQR Formula:

IQR = Q3 - Q1

In order you find Q3 and Q1, please follow these steps:

1. First, you need to order the list of numbers from least to greatest:

2, 6, 9, 10, 16, 28, 33

2. Then, you need to find the median, or the middle number.

2, 6, 9, 10, 16, 28, 33

3. In order to find IQR, you must find the first and third quartiles.

2, 6, 9, 10, 16, 28, 33

Q1 = 6

Q3 = 28

This is because 6 basically means that all the numbers leading to 6 would account for 25% of the data while all the numbers leading to 28 would account for 75% of the data, hence why these are called quartiles.

Now since you have Q1 and Q3, you follow the formula.

28 - 6 = 22

IQR = 22

6 0

3 years ago