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A carpenter has boards of lengths 24, 36, and 42 inches that must be cut into smaller boards of equal length, with no scrap wood left over.


What is the longest length of boards he can cut?


A. 2 inches

B.6 inches

To minimize production cost, we need to find the dimensions of the package that will satisfy the volume equation and minimize the cost equation. Taking the partial derivatives of the cost equation with respect to the variables 'r' and 'h', and setting them equal to zero, we can find the critical points. Solving these equations, we find that the dimensions for the package that will minimize production cost are a height of 7 cm and a radius of 2.5 cm.

To find the dimensions of the package that will minimize production cost, we need to consider the cost of the sides and bottom made of styrofoam and the cost of the top made of glued paper. Let's denote the radius of the base of the cylinder as 'r' and the height of the cylinder as 'h'.

The volume of a cylinder is given by the formula V = πr²h. We know that the volume of the package needs to be 350 cubic centimeters, so we can write the equation as πr²h = 350.

The cost of the sides and bottom of the container is given by the formula C1 = 0.03 * (2πrh + πr²), and the cost of the top is given by the formula C2 = 0.08 * πr².

To minimize the cost, we need to find the values of 'r' and 'h' that satisfy the volume equation and minimize the cost equation.

Since this is a calculus problem, we will differentiate the cost equation with respect to 'r' and 'h', and set the derivatives equal to zero to find the critical points.

Taking the partial derivatives of C1 and C2 with respect to 'r' and 'h', we get:

Solving these equations, we find that h = 7 cm and r = 2.5 cm. Therefore, the dimensions for the package that will minimize production cost are a height of 7 cm and a radius of 2.5 cm.

Answer:

A=28.25 is the rounded form of the answer.

A = 28.245673384359 in^2

Answer:

Area: 28.25

Step-by-step explanation:

Area = Circumference^2

4 (pi)

18.84^2 = 354.9456

4(pi) = 12.5663706144

354.9456 ÷ 12.5663706144 = 28.2456733843

Rounded to the nearest hundreth, the 5 in the thousandths place converts the 4 in the hundreths place into 5.

Answer: 275

Step-by-step explanation:

Let the total number of hamburger be x and the total number of cheeseburgers be y. That is

x + y = 500 ................... equation 1

Since there were 50 fewer cheeseburgers sold than hamburgers, this means that the number of hamburgers sold is more than cheeseburgers by 50 , that is

x - y = 50 ............. equation 2

solving the resulting linear equation by substitution method;

From equation 2 , make x the subject of the formula , that is

x = 50 + y ................ equation 3

substitute equation 3 into equation 1 , we have

50 + y + y = 500

50 + 2y = 500

2y = 500 - 50

2y = 450

y = 450/2

y = 225

substitute y = 225 into equation 3 to find the value of x. we have

x = 50 + y

x = 50 + 225

x = 275

Therefore 275 hamburgers were sold

Answer:

y = 2 over 3 x + 5

Step-by-step explanation:

m= y2-y1/x2-x1

m= (9-7)/(6-3)

m=2/3

y=mx+b

y= 2/3x+b

(3,7)

7=2/3(3)+b

7=6/3+b

7=2+b

5=b

b=5

y=2/3x+5

Step-by-step explanation:

(3)(7)(13) = 273

(3)(3)(7)(13) = 819

(3)(7)(7)(13) = 1911

(3)(7)(13)(13) = 3549

(3)(3)(3)(7)(13) = 2457

(3)(7)(7)(7)(13) = 13377

(3)(7)(13)(13)(13) = 46137

....

Answer:

8(3+4x)

Step-by-step explanation:

it it can be pulled out of both 24 and 32.

Answer:

12x + 18y

Step-by-step explanation:

Lets first write the quantity "two x plus three y", this means, that we have x and y as unknowns. x is 2 times, so we have 2x; and y is 3 times, so he have 3y:

2x + 3y

Additionally we have that this quantity is six times, what means that it is multiplied by a number 6:

6(2x +3y)

If you like, we can apply the distributive property that means that given any three numbers a, b, c we have that a(b+c)=ab + ac. In our case a is 6, b is 2x and c is 3y. Thus, operating:

6(2x +3y) = 6*2x + 6*3y = 12x + 18y

So, "six times the quantity two x plus three y" is 12x + 18y

Answer:

[tex]6.5 \times 10^{-9}[/tex]

Step-by-step explanation:

The scientific notation of a number is to express the number in the form [tex]x\times 10^{y}[/tex] where x is the base of ten.

So, if we want to express the number 0.0000000065, then we can write it as

[tex]\frac{6.5}{1000000000} = 6.5 \times 10^{-9}[/tex].

So, we can write 0.0000000065 as [tex]65\times 10^{-10}[/tex] or [tex]0.65 \times 10^{-8}[/tex] but those are not the scientific notation.  

Therefore, [tex]6.5 \times 10^{-9}[/tex] is the scientific notation of 0.0000000065. (Answer)

Answer:

X= 0

Step-by-step explanation:

Answer: 7.) Wrong Question

8.) 8

(9.) - 2

Step-by-step explanation:

7.) 10n - 11 = 3 + 4n + 6n

Collect like terms

10n - 4n - 6n = 3 + 11

10n - 10n = 14

It's not possible to have zero n = 14

8.) -8(k + 4) = - 96

Exland the bracket

-8k - 32 = - 96

Collect like terms

-8k = - 96 + 32

-8k = - 64

Divide through by -8

-8k/-8 = - 64/-8

k = 8

9.) -6(2k + 5) + 7k = -6 + 7k

Open the bracket

-12k - 30 +7k = -6 + 7k

Collect like terms

-12k + 7k - 7k = - 6 + 30

-12k = 24

Divide both sides by -12

-12k/-12 = 24/-12

k = -2

Answer:  x = -13

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

Work Shown:

19 = 38/2

19 & 1/2 = 19 + 1/2

19 & 1/2 = 38/2 + 1/2

19 & 1/2 = (38+1)/2

19 & 1/2 = 39/2

-------

In the original equation, replace the "19 & 1/2" with "39/2" to get

(5/2)x + (1/2)x = 39/2 + (9/2)x

Each term involves a fraction with a denominator of 2. If we multiply both sides by the common denominator, then the fractions will cancel out

For example: 2*(5/2)x = (10/2)x = 5x

So after multiplying both sides by 2, you should have this new equation

5x + 1x = 39 + 9x

--------

Let's solve that latest equation for x

5x + 1x = 39 + 9x

6x = 39 + 9x

6x-9x = 39 + 9x-9x ... subtract 9x from both sides

-3x = 39

-3x/(-3) = 39/(-3) .... divide both sides by -3

x = -13

Answer:

so if all of the awnsers are whole numbers it is going to be 12/12

Step-by-step explanation:

Answer:

12

Step-by-step explanation:

Unit rate is a rate that has 1 as denominator.

Given the rate: 6 inches / 12 hours, we have to divide numerator and denominator by 12 to transform the denominator in 1, as follows:

[tex]\frac{6}{12} = \frac{6/12}{12/12} =[/tex] 0.5 inches/hour

Answer:

The answer is yes :)

Step-by-step explanation:

Answer:

66.67%

Step-by-step explanation:

Marie's earnings at a new job : $15 per hour

Her earnings during training : $10 per hour

percentage of Marie's regular rate earned by her during training

                 = [tex]\frac{Her\ earnings\ during\ training}{Her earning\ at\ a\ new\ job}[/tex]×100

                 = [tex]\frac{10}{15}[/tex]×100

                 = [tex]\frac{200}{3}[/tex]

                 = 66.67%

∴percentage of Marie's regular rate earned by her during training = 66.67%

Final answer:

To find what percentage of Marie's regular rate she will earn during training, divide her training rate by her regular rate and multiply by 100%.

Explanation:

Marie will earn $10 per hour during training. Her regular rate is $15 per hour. To calculate what percentage of her regular rate she will earn during training, we use the formula: percentage = (training rate / regular rate) * 100%. Plugging in the values, we get (10 / 15) * 100% = 66.67%. Therefore, Marie will earn 66.67% of her regular rate during training.

Answer:

Maura runs faster

Step-by-step explanation:

To avoid decimals ,multiply all by 100

Maura 500 mi/125h = 4 mi/h

Ava 300 mi /83 h =3.61445.....mi/h

Answer:

still from 2019

wsp my dude

Step-by-step explanation:

Answer: 2/6 or 1/3

Step-by-step explanation:

2/3 = 4/6. 1 /6 Two times is 2/6 so by subtracting you should get 2/6 or 1/3

Answer:

[tex]x=16[/tex]

 Lenght of the equal sides: [tex]71\ units[/tex]

Step-by-step explanation:

You can observe in the picture that the triangle have two angles that have equal measure; therefore, it is an Isosceles triangle.

By definition, Isosceles triangles have two equal angles and two equal sides.

Based on this, you can write the following equation:

[tex]9x-73=3x+23[/tex]

Solving for "x", you get this value:

[tex]9x-3x=23+73\\\\6x=96\\\\x=\frac{96}{6}\\\\x=16[/tex]

Finally, you can subsitute the value of "x" into any expression to find the lenght of the equal sides:

[tex]3(16)+23=71\ units[/tex]

The solution on the graph is the coordinate of the point where both lines intersect

Given the equation of two lines expessed as;

y = -3x + 3 and y = 12 x - 4

Since both expressions ara y values, equating both will give;

-3x + 3 = 12x - 4

-3x - 12x = -4 - 3

-15x = -7

x = 7/15

Substitute x = 7/15 into any of the equatons;

y = -3x + 3

y = -3(7/15) + 3

y = -7/5 + 3

y = 8/5

Hence the solution as an ordered pair is (7/15, 8/15)

The solution on the graph is the coordinate of the point where both lines intersect as shown in the graph below;

Learn more on graphs here: https://brainly.com/question/18872003

Answer:

slope is 1 and y-intercept is 3

Step-by-step explanation:

equation is in the format y=mx+c

c = y-intercept

m = slope

if there is no  number in front of x it is assumed that 1 is in front of x

Answer:

The slope is 1 and the y-intercept is 3

Step-by-step explanation:

y = x+3 has the form y = mx + b.  If we compare these two equations, it becomes readily apparent that m = 1 and b = 3.

The slope is 1 and the y-intercept is 3 (or    (0, 3)     )

Answer:

$96,000

Step-by-step explanation:

To find Jen's annual salary, divide Bill's annual salary of $64,000 by 2/3, resulting in Jen's salary being $96,000.

To calculate Jen's annual salary based on the information that Bill's annual salary is 2/3 of Jen's, first express Bill's salary as a fraction of Jen's:

Therefore, Jen's annual salary is $96,000.

Answer:

14 +(9×x) = y x being the number of pizzas that are not the first one and y is the amount it all cost

Answer:

See explanation below:

Step-by-step explanation:

Here we want to express the number of pizza delivered. Lets try to understand the case.

If we buy one pizza (p1) the cost is 14 because of the delivery plus 9 for the cost of the pizza. So:

p1 = 14 + 9 = 23

If we buy 2 pizzas (p2) we have to pay 14 for the delivery and 9 for each pizza:

p2 = 14 + 9 + 9 = 14 + 2*9 = 32

If 3 pizzas are bought (p3) the reasoning is the same, 14 for the delivery and 9 for each pizza:

p3 = 14 + 9 + 9 + 9 = 14 + 3*9 = 41

Here we can see a pattern: for any amount of pizzas n the cost will be 14 plus n-times 9, the price of a pizza. It is:

pn = 14 + 9*n

But we can separate the last equation in such a way:

pn = 14 + 9*(n-1) + 9 , as 9*(n-1) + 9 = 9*n

But if our general formula it true, the first to terms on the right are the vost of buying n-1 pizzas, so:

pn = pn-1 + 9

We can keep on separating terms as we did to arrive to this form, this is, separating the 9n in  9*(n-2) + 9*2,  9*(n-3) + 9*3, ...,  9*(n-m) + 9*m for any m < n . Here I will keep pn = pn-1 + 9 as the Recursive Formula, but you could use other formulation with, e.g., pn-2, pn-3, etc, using the above recursion.

Now, the explicit formula is such that gives as the value of any amount, finding all the terms of the sequence with those not depending on previous terms (contrary to the recursive). This formula was showed above and is: pn = 14 + 9*n. This is, given any number of pizzas we can find the cost without knowing the cost of previous values.

So, we have:

Recursive: pn = pn-1 + 9

Explicit: pn = 14 + 9*n

7.

y + 8x = -2

f(x) = y = -8x - 2

f(0)=-2, f(1)=-10, f(2)=-18, f(-2)=14

8.

12x + 3y = 15

3y = -12x + 15

f(x) = y = -4x + 5

f(-1)=9, f(0)=5, f(1)=1

9.

y - 3x = 9

f(x) = y = 3x + 9

f(-2)=3, f(0)=9, f(2)=15

10.

0.4y + 2x = 1.2

4y + 20x = 12

4y = -20x + 12

f(x) = y = -5x + 3

f(-3) = 18, f(0)=3

Answer:

A. 2- i

Step-by-step explanation:

Answer:

2 + i

Step-by-step explanation:

Answer:

h=24

Step-by-step explanation:

h/2+3=3(h/4-1)

h/2+3=3h/4-3

h/2-3h/4=-3-3

2h/4-3h/4=-6

-h/4=-6

-h=-6*4

-h=-24

h=24

Answer:

Step-by-step explanation:The sum of three consecutive odd numbers is 183. Find the numbers. 57; 59; 67 59; 61; 63 60; 61; 62 65; 67; 69.

A graph of the line with slope 2/5 and passing through the point (-2, 4) is shown on the coordinate plane in the image below.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

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

Where:

At data point (-2, 4) and a slope of 2/5, an equation for this line can be calculated by using the point-slope form as follows:

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

y -  4 = 2/5(x - (-2))  

y -  4 = 2/5(x + 2)

y = 2/5(x) + 4/5 + 4

y = 2/5(x) + 24/5

Read more on point-slope here: brainly.com/question/24907633

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Answer:

The exact volume of a cone is 10 yd³.

Step-by-step explanation:

Given:

Height of cone (h) is 6 yd and base (b) is 5 yd.

Now, to get the volume we need to find the radius.

So, to find radius we put the formula of base:

Let the radius be [tex]r[/tex]

Base=[tex]\pi r^2[/tex]

[tex]b=\pi\times r^{2}[/tex]

[tex]5=\pi\times r^{2}[/tex]

Dividing both sides by π and using square root on both the sides:

[tex]5\div \pi=(\pi \times r^{2}) \div \pi[/tex]

[tex]\frac{5}{\pi}=r^{2}[/tex]

[tex]\frac{5}{\pi}=r^{2}[/tex]

[tex]r^{2}=\frac{5}{\pi}-----1[/tex]

Now, putting the formula for getting the volume (v) of cone:

Volume = [tex]\pi r^2\frac{h}{3}[/tex]

[tex]v=\pi\times r^{2}\times \frac{6}{3}[/tex]

[tex]v=\pi\times r^{2}\times 2[/tex]

Substituting the value of [tex]r^2[/tex] from equation (1), we get:

[tex]v=\pi\times \frac{5}{\pi}\times 2[/tex]

[tex]v=10[/tex] yd³

Volume of cone is 10 yd³

Therefore, the exact volume of a cone is 10 yd³.

Answer:

No Solution.

Step-by-step explanation:

3x-7=3x+1

3x-3x-7=1

0-7=1

-7=1

no solution

y = 2x

Step-by-step explanation:

The linear functions are given by:

y = mx+b

To find the slope, we have to find the change in x and y

The change in x is:

[tex]-14-(42) =-14+42 =28\\14+14= 28[/tex]

The change in y is:

[tex]-28-(-84) = -28+84 = 56\\28-(-28) = 28+28 = 56[/tex]

The slope is:

[tex]Slope = \frac{change\ in\ y}{Change\ in\ x}\\=\frac{56}{28}\\= 2[/tex]

Putting slope in equation of linear function

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

To find the value of b, we will put any pair of input and output in the slope-intercept form

So, putting (14,28)

[tex]28 = 2(14) + b\\28 = 28+b\\b = 28-28\\b = 0[/tex]

Hence,

the linear equation for function is:

y = 2x

Keywords: Linear function, slope intercept form

Learn more about linear function at:

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