Find the extreme values of the function f(x, y) = 4x + 8y subject to the constraint g(x, y) = x2 + y2 − 5 = 0.

Answers

Answer 1
[tex]L(x,y,\lambda)=4x+8y+\lambda(x^2+y^2-5)[/tex]

[tex]L_x=4+2\lambda x=0[/tex]
[tex]L_y=8+2\lambda y=0[/tex]
[tex]L_\lambda=x^2+y^2-5=0[/tex]

[tex]L_y-2L_x=2\lambda y-4\lambda x=2\lambda(y-2x)=0\implies y=2x[/tex]

[tex]xL_x+yL_y=2\lambda(x^2+y^2)+4x+8y=10\lambda+4x+8y=0\iff5\lambda+2x+4y=0[/tex]

[tex]y=2x\implies 5\lambda+5y=0\implies 5(\lambda+y)=0\implies y=-\lambda[/tex]

[tex]4+2\lambda x=0\iff4+\lambda y=0\iff4-y^2=0\implies y=\pm2[/tex]
[tex]y=\pm2\implies x=\dfrac y2=\pm1[/tex]

In other words, there are two critical points, [tex](1,2)[/tex] and [tex](-1,-2)[/tex]. At the first point, there is a maximum of [tex]f(1,2)=20[/tex] and at the second, there is a minimum of [tex]f(-1,-2)=-20[/tex].

Related Questions

Glenda began the day with a golf score of -6 and ended with a score of -10. Which statement represents her golf score for that day?

A) -6 - ( - 10) = 4

B) - 10 - ( - 6) = - 4

C) - 6 + ( - 10) = - 16

D) - 10 + ( - 6) = - 16

Answers

B)
She started with -6 and ended up with -10 so her score was 4 less than what she started with so it is -4
B. I need 20 characters to answer so I hope you have a wonderful day

Two redwood trees are the same height. A third tree's height is x feet. The first tree is 50 feet more than three times the height of the third tree. The second tree is 100 feet taller than twice the height of the third tree. What is the height, in feet, of the third tree? A. 150 B. 50 C. 200 D. 100

Answers

3rd tree = x

1st tree = 3x+50

2nd tree = 2x+100


3x+50 = 2x+100

subtract 50 from each side

3x= 2x+50

subtract 2x from each side

x = 50

 so 3rd tree = 50 feet tall

 Answer is B

Cody saves all his nickles. Today he is getting them out of his piggy bank and wrapping them to take to the bank. He finds he has 360 nickels. It takes 40 Nicole's to fill each paper wrapper and make a roll. How many wrappers does he need? Explain how you solved this problem and how you know your answer is correct.

Answers

Simple explanation:
Divide 360 by 40
360/40 = 9
Cody needs 9 wrappers.

3rd grader explanation:
Explain that Cody wants to divide 360 into __ groups of 40.
Use an illustration of separating things into equal groups (you can simply use separating 6 objects into equal groups of 2).
Explain that subtraction, multiplication, or addition will not help.
Tell him/her to use division.
Although 360 and 40 are large numbers, explain that the 0's can be removed into 4 and 36.
36 / 4 = 9
Since the answer is 9, Cody needs 9 wrappers.

Have an awesome day! :)
350 divided by 40 equals 8.75. He needs 9 because he fills 8 completely but has some left over. He can't just not fill them, so you would need an extra. 9 is the answer.

what is the answer 4(6/10−1/5)+0.4=

Answers

For this kind of problem it'd be better to ask, "how would we evaluate this?
The task here is EVALUATION:  Finding the value of a mathematical expression.

Use Order of Operations rules.  Often dubbed "PEMDAS," these rules start by emphasizing that anything enclosed inside parentheses MUST be done first.  Next come mult. and di.  Last are add'n and subt'n.

In 4(6/10−1/5)+0.4, we MUST combine 6/10 and -1/5 FIRST.

That's the same as 3/5-1/5, or 2/5.  Then the problem looks like:

Evaluate:  4(2/5)+0.4

Now, we MUST multiply before adding or subtracting.  We get

8/5 + 4/10, or 16/10+4/10 = 20/10, or 2 (answer).

Or in shorter answers 2

Bill worked 24 hours last week and earns $8.25 per hour. Use front-end estimation to find an estimate of the total amount of money he earned.

Answers

Answer:

The total amount of money he earned was $160

Step-by-step explanation:

Bill worked 24 hours last week

He earns $ 8.25 per hour.

We have to use front-end estimation, we have to round to the first number all the values. So. Bill worked h = 20 hours and he earns 3 =$8 per hour.

Total amount (T) wil be the hours (h) multiplied by the earning (e)

T = h*e

T = (20 hours)(8 $/hours)

T = 160 $

After using front-end estimation, we can conclude that Bil earned $160.

labored 24 hours remaining week

He earns $ 8.25 in step per hour.

We should use the front-stop estimation, we have to round to the first number of all of the values. So. invoice worked h = 20 hours and he earns 3 =$eight according to an hour.

the entire amount (T) can be the hours (h) extended through the earning (e)

T = (20 hours)(eight $/hours)

T = 160 $

After using front-end estimation, we will conclude that Bil earned $one hundred sixty.

How do you calculate front-end estimation?

front-cease estimation is a particular way of rounding numbers to estimate sums and variations. to apply the front-end estimation, add or subtract the handiest of the numbers in the finest location cost. Then upload the decimals rounded to the closest 10th.

what are front-end estimation examples?

In the front-cease estimation, we replace the original quantity with quite a number it is close by but the handiest has one non-zero digit. as an example, 20, three hundred, -50, 1,000, eighty,000. Take the quantity 32; the selections to update it with are 30 or forty (they both have the handiest one non-0 digit).

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I need help! Rewrite each product below using Distributive property.
a: 18(26)
b: 6(3405)
c: 21(35)
Thanks!!!p

Answers

Hi there,
a. 18(26) →18 × 26 = 468
b. 6(3405) → 6 × 3405 = 20,430
c. 21(35) → 21 × 35 = 735

The numbers in bold are the answers. Hope this helps and have a phenomenal day!

By using distributive property, the answers are :(a)468, (b) 20430, (c) 735.

Explain, distributive property.

Distributive property is the mathematical property to solve expressions in the form of a(b + c), in which firstly bracket term is solve and then other terms are solved.

(a)

The given term is,

18(26),

According to distributive property, bracket changes in multiply.

Therefore, 18(26)⇒18×26=468.

(b)

The given term is,

6(3405),

According to distributive property, bracket changes in multiply.

Therefore, 6(3405)⇒6×3405=20430.

(c)

The given term is,

21(35),

According to distributive property, bracket changes in multiply.

Therefore, 21(35)⇒21×35=735.

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Which statement best describes one of these transformations? Triangle 1 is rotated to result in triangle 2. Triangle 1 is dilated to result in triangle 3. Triangle 1 is reflected to result in triangle 4. Triangle 1 is stretched to result in triangle 5.

Answers

Answer:

Triangle 1 is rotated to result in triangle 2.

Step-by-step explanation:

Answer:

A

Step-by-step explanation:

GOT IT RIGHT

A gas tank has a leak. Each hour it loses 3 gallons of gas. Write the equation that represents how much gas, g , the tank loses over h hours.

Answers

Lost gas = g(h) = -(3 gallons / hour)h.

The rate of change, or the slope, of this relationship is -3 gal / hr

Find 14+1320
. Write your answer as a fraction in simplest form.

Answers

it is 7/660
hope this helps
it is           9             because                   1                 13                  9
            -----------                                --------------+---------------- =-----------------       
                 10                                             4                 20                 10
                                                                                                   

An angle measures 54 less than the measure of a complementary angle. What is the measure of each angle?

Answers

Complementary angles add up to 90 degrees. The first angle is "x" and the second is "x-54". So you add them up and get it equal to 90. Like 2x-54=90, them you solve and you will get x=72. That is the first angle which is 72 degrees. Then to get the second angle you take 90 and subtract 72 and get 18 degrees which is the second angle.

A sample of n = 9 scores has a variance of s2 = 36. what is the estimated standard error for the sample mean? 1 2 4 16

Answers

Given:
σ² = 36, the variance
n = 9, the sample size.

The standard deviation is 
σ = 6

By definition, the standard error is
σ/√n = 6/3 = 2

Answer:  2



Which number is a factor of 20 and also a prime number?


A.
4


B.
5


C.
7


D.


10

Answers

20 = 2 x 2 x 5, which can also be written 20 = 2² x 5
thus B is your Answer.
The answer is 5 because, 20= 5*4 then 4= 2*2, from these options 5 is the answer

A four person relay team completed a race in 72.4 seconds. On average what was each runner's time?

Answers

each runners time would be an average of 18.1
72.4/4=18.1

Answer:

i believe it's 18.1

Step-by-step explanation:

WILL GIVE BRAINEST
Which system of inequalities is shown?

Answers

The answer should be C since you first have to find the equation of the 2 lines, which is y=x and y=4. Then, look to see if the shaded part's above the lines. It is for both, so y>x and y>4, which is C.

The system of inequalities are y > x and y > 4 option (C) y > x, y > 4 is correct.

What is inequality?

It is defined as the expression in mathematics in which both sides are not equal they have mathematical signs either less than or greater than known as inequality.

We have a graph shown in the picture,

As we can see in the picture there two lines are shown which represent the inequality and the shaded region is the solution to the pair of the linear inequality.

y > x and

y > 4

Thus, the system of inequalities are y > x and y > 4 option (C) y > x, y > 4 is correct.

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Evaluate the line integral ∫cx4zds, where c is the line segment from (0,6,5) to (8,4,1).

Answers

Parameterize the path [tex]\mathcal C[/tex] by

[tex]\mathbf r(t)=\langle0,6,5\rangle(1-t)+\langle8,4,1\rangle t=\langle8t,6-2t,5-4t\rangle[/tex]

with [tex]0\le t\le1[/tex]. Then

[tex]\displaystyle\int_{\mathcal C}x^4z\,\mathrm dS=\int_{t=0}^{t=1}(8t)^4(5-4t)\sqrt{8^2+(-2)^2+(-4)^2}\,\mathrm dt=8192\sqrt{21}-16384\sqrt{\frac73}[/tex]
Final answer:

To evaluate the line integral, we need to parameterize the line segment and calculate the differential arc length. After simplifying the integrand, we can evaluate the line integral by integrating the simplified expression.

Explanation:

To evaluate the line integral ∫cx^4zds, where c is the line segment from (0,6,5) to (8,4,1), we need to parameterize the line segment c. Let's parameterize it with t, where t varies from 0 to 1. The parameterization of c is given by:

x(t) = 8t, y(t) = 6 - 2t, z(t) = 5 - 4t.

Next, we need to calculate ds, which is the differential arc length along the line. The differential arc length ds is given by:

ds = √(dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2 dt.

Substituting the parameterized values of x(t), y(t), and z(t) into the above equation, we get:

ds = √[(8)^2 + (-2)^2 + (-4)^2] dt = √84 dt.

Now, we can rewrite the line integral as:

∫cx^4zds = ∫(8t)^4(5 - 4t)√84 dt.

Next, we simplify the integrand:

(8t)^4(5 - 4t) = 4096t^4(5 - 4t).

Finally, we evaluate the line integral by integrating the simplified integrand over the interval [0, 1]:

∫(8t)^4(5 - 4t)√84 dt = √84 ∫4096t^4(5 - 4t) dt.

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A shop advertised a 22%-off sale. If a coat originally sold for ​$257​, find the decrease in price and the sale price

Answers

Decrease in price is $56.54. The new sale price is $200.46

20,880 times 2/5 worked out

Answers

20,880 x 2/5 

= (20,880 x 2)/5 

= (41,760)/5 

= 8,352 

The multiplication [tex]20880 \times \dfrac{2}{5}[/tex] is [tex]\boxed{8352}.[/tex]

Further explanation:

Always use the PEDMAS rule to solve the grouping of multiplication, addition, subtraction and division.

Here, P is parenthesis, E is exponents, M is multiplication, D is division, A is addition and S is subtraction.

Given:

The expression is [tex]20880 \times \dfrac{2}{5}.[/tex]

Explanation:

The given expression is [tex]20880 \times \dfrac{2}{5}.[/tex]

Consider the expression [tex]20880 \times \dfrac{2}{5}[/tex] as A.

[tex]A = 20880 \times \dfrac{2}{5}[/tex]

Solve the above expression to obtain the simplest form.

[tex]\begin{aligned}A&= 20880 \times \frac{2}{5}\\&= \frac{{41760}}{5}\\&= 8352\\\end{aligned}[/tex]

Another way of solving the expression can be expressed as follows,

First divide 2 by 5.

[tex]\begin{aligned}b&= \frac{2}{5}\\&= 0.4\\\end{aligned}[/tex]

Now multiply 20880 to 0.4.

[tex]\begin{aligned}A&= 20880 \times 0.4\\&= 8352\\\end{aligned}[/tex]

The multiplication [tex]20880[/tex] times [tex]\dfrac{2}{5} \:{\text{is}\: \boxed{8352}.[/tex]

Learn more

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Answer details:

Grade: High school

Subject: Mathematics

Chapter: Number system

Keywords: factor, factorization, 20880, 2/5, times, greatest common factor, groups, multiplication, product, identities, common factor, expression, terms, grouping.

joe is using his allowance money to download songs from the internet. he can download them at a rate of eight songs every 10 minutes. Jasmine, who is downloading songs with the money she made from her part time job, can download at a rate of 12 songs every 15 minutes. Who is downloading songs faster?

Answers

Joe: 8 songs in 10 minutes = 8/10 = 0.8 songs per minute

Jasmine: 12 songs in 15 minutes = 12/15 = 0.8 songs per minute

 they are downloading at the same speed

If p(a) = 0.65, p(b) = 0.58, and p(a and
b.= 0.76, then p(a or
b.is:

Answers

i think the answer to  b is 0.47

The value of P( a or b)= 0.47.

what is Conditional Probability?

The potential of an event or outcome occurring based on the existence of a prior event or outcome is known as conditional probability. It is determined by multiplying the likelihood of the earlier occurrence by the increased likelihood of the later, or conditional, event.

Conditional probability is the likelihood that any event A will occur in the presence of another event B that is related to A. P(A|B) illustrates it.

Given:

p(a) = 0.65, p(b) = 0.58, and p(a and b)= 0.76

Now, using the formula

P(a∪b) = P(a) + P(b) - P(a∩b)

           = 0.65 + 0.58 - 0.76

           = 0.47

Hence, the value of P( a or b)= 0.47.

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I suck at word problems help

Answers

$112
hope this helped

Graph the system of inequalities. Then use your graph to identify the point that represents a solution to the system. x < 2 and y ≤ 4x + 1


(5, 6)


(5, –2)

(–3, –15)


(1, 11)

Answers

We want to solve the inequalities
x < 2
y ≤ 4x + 1

Graph the lines:
x = 2
y = 4x +1

The solution to the system of inequalities is the shaded region shown in the graph below.

Test the given points to see whether they fall within the solution region.

Point   Pass/Fail
--------  --------------
(5,6)     Fail
(5,-2)    Fail
(-3,-15)  Pass
(1,11)       Fail

Answer: (=3, -15)

On your last credit card statement, you had a previous balance of $98.14, payments of $10, a periodic rate of 1.34 percent, and new charges totaling $34.89. What is your new balance?

A. $88.14

B. $100.58

C. $124.21

D. $189.35

Answers

The correct answer is D

Doomtown is 300 miles due west of Sagebrush and Joshua is due west of Doomtown. At 9 a.m. Mr. Archer leaves Sagebrush for Joshua. At 1 p.m. Mr. Sassoon leaves Doomtown for Joshua. If Mr. Archer travels at an average speed 20 mph faster than Mr. Sassoon and they each reach Joshua at 4 p.m., how fast is each traveling?

Answers

Refer to the diagram shown below.

Let the distance that Sassoon travels is d miles.
Let his average speed be x mph.
The time of travel is from 1 p.m. to 4 p.m., which is 3 hours.
Therefore 
3x = d                            (1)

The distance that Archer travels is (d + 300) miles.
The time of travel is from 9 a.m. to 4 p.m., which is 7 hours.
The average traveling speed is (x + 20) mph, therefore
7(x + 20) = d + 300     
That is,
7x + 140 = d + 300
7x = d + 160                  (2)

Subtract (1) from (2).
7x - 3x = d + 160 - d
4x = 160
x = 40 mph  (Sassoon's average speed)
x+20 = 60 mph (Archer's average speed)
From (1), obtain d = 120 mi

Answer:
Archer's speed is 60 mph.
Sassoon's speed is 40 mph.

Answer:Archer's speed is 60 mph.

Sassoon's speed is 40 mph.

Step-by-step explanation:

Let the distance that Sassoon travels is d miles.

Let his average speed be x mph.

The time of travel is from 1 p.m. to 4 p.m., which is 3 hours.

Therefore

3x = d                            (1)

The distance that Archer travels is (d + 300) miles.

The time of travel is from 9 a.m. to 4 p.m., which is 7 hours.

The average traveling speed is (x + 20) mph, therefore

7(x + 20) = d + 300    

That is,

7x + 140 = d + 300

7x = d + 160                  (2)

Subtract (1) from (2).

7x - 3x = d + 160 - d

4x = 160

x = 40 mph  (Sassoon's average speed)

x+20 = 60 mph (Archer's average speed)

From (1), obtain d = 120 mi

Answer:

Archer's speed is 60 mph.

Sassoon's speed is 40 mph.

Which situation could best be represented by this linear equation?

2.75x + 3.25y = 215

Answers

2.75x + 3.25y = 215

answer is the last one (D)

Answer:

The correct option 4.

Step-by-step explanation:

The given equation is

[tex]2.75x+3.25y=215[/tex]          .... (1)

Let x is the number of pretzels sold and y is the number of popcorn bags sold.

Total sales is defined as

[tex]Sales=C_1x+C_2y[/tex]         ... (2)

From (1) and (2) it is clear that,

The cost of each pretzel is $2.75 or 275 cents. The cost of each popcorn bag is $3.25 or 325 cents ($1 = 100 cents). Total sales is $215.

[tex]325-275=50[/tex]

Pretzels cost is 50 cents less than popcorn bags. The variable x represents the number of pretzels sold and y represents the number of popcorn bags sold. The total sales was $215.

Therefore the correct option 4.

Evaluate the triple integral of f(x,y,z)=z(x2+y2+z2)−3/2f(x,y,z)=z(x2+y2+z2)−3/2 over the part of the ball x2+y2+z2≤49x2+y2+z2≤49 defined by z≥3.5z≥3.5. ∫

Answers

I'm going to take a wild guess and assume you mean

[tex]f(x,y,z)=z(x^2+y^2+z^2)^{-3/2}=\dfrac z{(x^2+y^2+z^2)^{3/2}}[/tex]

(as opposed to subtracting 3/2, say)

Convert to spherical coordinates, taking

[tex]\begin{cases}x=\rho\cos\theta\sin\varphi\\y=\rho\sin\theta\sin\varphi\\z=\rho\cos\varphi\end{cases}\implies\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]

Now when [tex]z=\dfrac72[/tex], below the "cap" of the ball we have a right triangle with hypotenuse coinciding with the ball's radius and a leg of length [tex]\dfrac72[/tex], with the angle between them [tex]\varphi[/tex] satisfying

[tex]\cos\varphi=\dfrac{\frac72}7=\dfrac12\implies\varphi=\cos^{-1}\dfrac12=\dfrac\pi3[/tex]

Furthermore, when [tex]z=\dfrac72[/tex], we have

[tex]\dfrac72=\rho\cos\varphi\implies\rho=\dfrac72\sec\varphi[/tex]

Call the cap [tex]C[/tex]. Then the triple integral over [tex]C[/tex] is

[tex]\displaystyle\iiint_Cf(x,y,z)\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi/3}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=7/2\,\sec\varphi}^{\rho=7}\frac{\rho\cos\varphi}{(\rho^2)^{3/2}}\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi[/tex]
[tex]=\displaystyle2\pi\int_{\varphi=0}^{\varphi=\pi/3}\int_{\rho=7/2\,\sec\varphi}^{\rho=7}\cos\varphi\sin\varphi\,\mathrm d\rho\,\mathrm d\varphi[/tex]
[tex]=\displaystyle\pi\int_{\varphi=0}^{\varphi=\pi/3}\sin2\varphi\left(7-\dfrac72\sec\varphi\right)\,\mathrm d\varphi[/tex]
[tex]=\displaystyle\frac{7\pi}2\int_{\varphi=0}^{\varphi=\pi/3}(2\sin2\varphi-\sin2\varphi\sec\varphi)\,\mathrm d\varphi=\frac{7\pi}4[/tex]
Final answer:

To evaluate a triple integral over a region of a sphere, we switch to spherical coordinates. We then express the integrand and the limits of the variables in terms of these new coordinates. Don't forget the Jacobian when changing variables.

Explanation:

The subject is a mathematical problem that requires evaluating a triple integral over a specific part of a ball in three-dimensional space. The function to integrate is f(x,y,z)=z*(x2+y2+z2)-3/2 and the region of integration is defined as the part of the sphere with radius 7 (x2+y2+z2<=49) in which z >= 3.5. Unfortunately, the solution to such a problem requires advanced mathematical skills and knowledge, such as spherical coordinates and integration techniques, which cannot be elaborated within the word limit here.

However, to solve such problems, you typically switch to spherical coordinates, where x=r*cos(theta)*sin(phi), y=r*sin(theta)*sin(phi), z=r*cos(phi). Then, you express your function and the limits of your variables r, theta, and phi in terms of these new coordinates. You need to consider the Jacobian determinant when changing variables. Eventually, you integrate over the specified region. The solution needs careful elaboration of every step due to the complexity of the integral.

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Which phrase describes the variable expression k -7? A) 7 less then k B) 7 more then k C) The quotient of k and -7 D) The product of k and -7

Answers

k - 7 = a) 7 less than k

hope this helps
The answer is 7 less than K

if you have 12 fish and 6 drown how many do you have left?

Answers

you will have 6 left
you have 12 fish left because fish don't drown

Find the length of the curve. r(t) = 3 i + t2 j + t3 k, 0 ≤ t ≤ 1

Answers

Call the curve [tex]C[/tex], then the length of [tex]C[/tex] is given by the line integral

[tex]\displaystyle\int_C\mathrm dS=\int_{t=0}^{t=1}\|\mathbf r'(t)\|\,\mathrm dt=\int_{t=0}^{t=1}t\sqrt{9t^2+4}\,\mathrm dt=\frac{13\sqrt{13}-8}{27}[/tex]
Final answer:

To determine the length of the curve, we find the derivative of the given vector function, compute its magnitude, and then compute the integral of the magnitude from 0 to 1. This involves calculus techniques.

Explanation:

To find the length of the curve, we have to compute the integral of the magnitude of the derivative on the interval from 0 to 1. First, we find the derivative of r(t) = 3 i + t^2 j + t^3 k which is r'(t) = 2t j + 3t^2 k. The magnitude of r'(t) is |r'(t)| = sqrt((0)^2 + (2t)^2 + (3t^2)^2) = sqrt(4t^2 + 9t^4). Thus the length of the curve is the integral of this magnitude from 0 to 1: ∫ from 0 to 1 of sqrt(4t^2 + 9t^4) dt. This integral can be evaluated using techniques from calculus.

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a moving company charges $250 to rent a truck and $0.40 for each mile driven. Mr. Lee paid a total of $314. Which equation can be used to find m, the number of miles he drove the moving truck?

Answers

The equation would be
314 = 0.4(x) + 250
Would you like me to explain?
Final answer:

To find the number of miles Mr. Lee drove the moving truck, we can set up an equation using the given information and solve for 'm'. The equation is $250 + $0.40x = $314, where 'x' represents the number of miles driven. Simplifying the equation, we find that Mr. Lee drove a total of 160 miles in the moving truck.

Explanation:

To find the number of miles Mr. Lee drove the moving truck, we can set up an equation using the given information. The moving company charges $250 to rent the truck and $0.40 for each mile driven. Let's let 'm' represent the number of miles he drove. The total cost Mr. Lee paid, including the rental fee and the cost per mile, is $314. So, the equation can be written as:

$250 + $0.40x = $314

To solve for 'm', we can subtract $250 from both sides of the equation:

$0.40x = $314 - $250

$0.40x = $64

Finally, we can divide both sides of the equation by $0.40 to find the value of 'm':

x = $64 / $0.40

x = 160

Therefore, Mr. Lee drove a total of 160 miles in the moving truck.

What are the values of a and b

Answers

the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are 8 and [tex]\( 2\sqrt{7} \)[/tex], respectively.

The image depicts a right-angled triangle with a hypotenuse of length 10 and one leg of length 6, with the other leg labeled as [tex]\( a \)[/tex]. There is also a smaller right-angled triangle within it, sharing the leg of length 6 with the larger triangle, with the hypotenuse of the smaller triangle labeled as 8 and the other leg labeled as [tex]\( b \).[/tex]

To find the lengths of [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse [tex](\( c \))[/tex] is equal to the sum of the squares of the lengths of the other two sides [tex](\( a \)[/tex] and [tex]\( b \)[/tex]): [tex]\( c^2 = a^2 + b^2 \).[/tex]

For the larger triangle:

[tex]\[ 10^2 = 6^2 + a^2 \][/tex]

For the smaller triangle:

[tex]\[ 8^2 = 6^2 + b^2 \][/tex]

We will solve these two equations to find [tex]\( a \)[/tex] and [tex]\( b \)[/tex]. Let's start with the calculations for [tex]\( a \)[/tex] and [tex]\( b \)[/tex] using the Pythagorean theorem.

Using the Pythagorean theorem for each triangle, we find:

For the larger triangle:

[tex]\[ a^2 = 10^2 - 6^2 \][/tex]

[tex]\[ a^2 = 100 - 36 \][/tex]

[tex]\[ a^2 = 64 \][/tex]

[tex]\[ a = 8 \][/tex]

For the smaller triangle:

[tex]\[ b^2 = 8^2 - 6^2 \][/tex]

[tex]\[ b^2 = 64 - 36 \][/tex]

[tex]\[ b^2 = 28 \][/tex]

[tex]\[ b = \sqrt{28} \][/tex]

[tex]\[ b = 2\sqrt{7} \][/tex]

So, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are 8 and [tex]\( 2\sqrt{7} \)[/tex], respectively.

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