Please help me with this

The equation of the tangent plane to the surface x=h(y,z) at the point (3,0,3) given that ∂h/∂y=−5 and ∂h/∂z=−3 is x = -5y -3z + 12.

In Mathematics, the equation for the tangent plane to a surface x = h(y,z) at a point can be determined by the following formula: x - x0 = ∂h/∂y(y - y0) + ∂h/∂z(z - z0). In your expression ∂h/∂y (derivatives of h with respect to y) is given as -5 and ∂h/∂z (derivatives of h with respect to z) is given as -3. By plugging the value of these derivatives and the point (3,0,3) into the formula, the equation of the tangent plane would be: x - 3 = -5(y - 0) - 3(z - 3).

After simplifying this, we'll get x = -5y -3z + 12. It's worth noting that derivation of the equation for a tangent plane to a parametrized surface relies heavily on partial derivatives and the study of multi-variable calculus.

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Answer:

$97.20

Step-by-step explanation:

Divide the tip amount $14.58 by the percentage given 15% or .15.

$14.58 ÷ .15 =$97.20

Check your work by multiplying the answer $97.20 by the percent of tip 15%. $97.20 × .15 =$14.58 (tip amount)

Answer:

C

Step-by-step explanation:

Answer:

A

Step-by-step explanation:

The volume (V) of a pyramid is calculated using the formula

V = [tex]\frac{1}{3}[/tex] × area of base × height

note the height = 48 × 10 = 480 ( 48 storeys at 10 feet )

V = [tex]\frac{1}{3 }[/tex] × 571,536 × 480

  = 91,445,760 ft³

[ 1 yd³ = 27 ft³ ], hence

V = [tex]\frac{91445760}{27}[/tex] = 3, 386, 880 yd³ → A

Answer:

Part A) The height of the cylinder is [tex]7.8\ in[/tex]

Part B) The volume of one tennis ball is [tex]9.2\ in^{3}[/tex]

Part C) The volume of the cylinder is [tex]41.4\ in^{3}[/tex]

Part D) The volume of space in the cylinder not taken by the tennis balls is [tex]13.8\ in^{3}[/tex]

Step-by-step explanation:

Part A) Each tennis ball has a diameter of 2.6 inches

What is the height of the cylinder?

we know that

The height of the cylinder is equal to the diameter of one ball of tennis multiplied by 3

so

[tex]h=2.6*3=7.8\ in[/tex]

Part B)  Find the volume of one tennis ball

The volume of the sphere (one tennis ball) is equal to

[tex]V=\frac{4}{3}\pi r^{3}[/tex]

we have

[tex]r=2.6/2=1.3\ in[/tex] ----> the radius is half the diameter

[tex]\pi=3.14[/tex]

substitute

[tex]V=\frac{4}{3}(3.14)(1.3)^{3}[/tex]

[tex]V=9.2\ in^{3}[/tex]

Part C) Find the volume of the cylinder

The volume of the cylinder is equal to

[tex]V=\pi r^{2} h[/tex]

we have

[tex]r=2.6/2=1.3\ in[/tex] ----> the radius is half the diameter

[tex]\pi=3.14[/tex]

[tex]h=2.6*3=7.8\ in[/tex]

substitute

[tex]V=(3.14)(1.3)^{2} (7.8)[/tex]

[tex]V=41.4\ in^{3}[/tex]

Part D) What is the volume of space in the cylinder not taken by the tennis balls?

we know that

The volume of space in the cylinder not taken by the tennis balls, is equal to the difference from the volume of the cylinder and the volume of three ball of tennis

[tex]V=41.4-(3)*(9.2)=13.8\ in^{3}[/tex]

Answer:

Part A) The height of the cylinder is

Part B) The volume of one tennis ball is

Part C) The volume of the cylinder is

Part D) The volume of space in the cylinder not taken by the tennis balls is

Step-by-step explanation:

Part A) Each tennis ball has a diameter of 2.6 inches

What is the height of the cylinder?

we know that

The height of the cylinder is equal to the diameter of one ball of tennis multiplied by 3

so

Part B)  Find the volume of one tennis ball

The volume of the sphere (one tennis ball) is equal to

we have

----> the radius is half the diameter

substitute

Part C) Find the volume of the cylinder

The volume of the cylinder is equal to

we have

----> the radius is half the diameter

substitute

Part D) What is the volume of space in the cylinder not taken by the tennis balls?

we know that

The volume of space in the cylinder not taken by the tennis balls, is equal to the difference from the volume of the cylinder and the volume of three ball of tennis

Step-by-step explanation:

Answer:

  D) [tex]\left[\begin{array}{c}\frac{5}{4}\\-\frac{1}{2}\end{array}\right][/tex]

Step-by-step explanation:

For matrix [tex]\left[\begin{array}{cc}a&b\\c&d\end{array}\right][/tex]

the inverse matrix is the transpose of the cofactor matrix, divided by the determinant: [tex]\dfrac{1}{ad-bc}\left[\begin{array}{cc}d&-b\\-c&a\end{array}\right][/tex]

Your inverse matrix is: [tex]\dfrac{1}{2(-3)-(1)(2)}\left[\begin{array}{cc}-3&-1\\-2&2\end{array}\right][/tex]

so the solution is ...

[tex]\left[\begin{array}{c}x\\y\end{array}\right]=\left[\begin{array}{cc}\frac{3}{8}&\frac{1}{8}\\\frac{1}{4}&-\frac{1}{4}\end{array}\right] \cdot\left[\begin{array}{c}2\\4\end{array}\right] =\left[\begin{array}{c}\frac{5}{4}\\-\frac{1}{2}\end{array}\right] \qquad\text{matches selection D}[/tex]

Answer:

Note: the last sentence in the problem statement text should read, "Therefore, the volume of the sphere is 4/3πr³ by Cavalieri's principle.

Step-by-step explanation:

I believe it can help a lot if you have seen and understand this derivation of the volume of a sphere. Here is the basic idea.

Shown in the attachment is a cross section of half the volume under consideration. Basically, it is showing one cone and the (red) top hemisphere in a (green) cylinder of radius R and height R. (The problem text refers to a sphere and two cones in a cylinder of height 2R. This is the top half of that geometry.) Actually, only the left edge of the cone is represented here, in order to avoid cluttering the diagram.

We can use this figure to think about a horizontal cross section (cut plane) of this geometry at height h from the center of the sphere. We want to consider the annulus of inner radius C between the cylinder of radius R and the cone, and we want to consider the circle of radius S where the cut plane intersects the hemisphere.

Because the cone has a height of R and a radius of R, the radius C of the cross section will be the same as the height h. That is, in our figure, h = C. We know from the Pythagorean theorem that ...

  h² + S² = R²

  S² = R² - C² . . . . . . subtract h² and substitute C for h

The area of the circular cross section of the hemisphere is πS², and the area of the annulus between the cylinder and cone is π(R² - C²). The above equation tells us these areas are the same.

By Cavalieri's principle, since the cross sections have the same area at every height, the volume of the space between the cylinder and cone is the same as the volume of the hemisphere. Using the formulas for the volumes of cylinder and cone, we find the difference to be ...

  difference volume = hemisphere volume = πr²·r - 1/3πr²·r = 2/3πr³

__

Once this approach to the sphere volume formula derivation is understood, filling in the blanks in your problem statement may become much simpler.

Answer:

25%

Step-by-step explanation:

Discount = Original Price x Discount %/100

Discount = 21.89 × 25/100

Discount = 547.25/100

You save = $5.4725

Final Price = Original Price - Discount

Final Price = 21.89 - 5.4725

Final Price = $16.4175

Answer:

25% off

Step-by-step explanation:

Answer:

D. Hot tub

Explanation:

1000 L is equal to about 246 gallons and even a deep bathtub couldn't hold that much water.

p.s. Loving your username

Answer:

the area of the hexagon is approx. 187.1 in²

Step-by-step explanation:

Picture this regular polygon as being a hexagon made up of six equilateral triangles of side 12 in.  We find the area of one such triangle and then multiply that by 6 to obtain the total area of the hexagon.

One such equilateral triangle has three sides all of length 12 in, and all the interior angles are 60°.  The height of one such triangle is

h = (12 in)sin 60°, or

                  √3

h = (12 in) -------- = 6√3 in

                     2

So, with base 12 in and height 6√3 in, the area of one such equilateral triangle is

A = (1/2)(12 in)(6√3 in) = 36√3 in²

and the total area of the hexagon is 6(36)√3 in², or approx. 187.1 in²

Answer:

B. {(4,6), (8,8), (–3,6)}

Step-by-step explanation:

The given inequality is

[tex]y\ge\frac{1}{4}x+5[/tex]

The set of points that satisfy the inequality are solutions.

We can verify that,all the points in the set  {(4,6), (8,8), (–3,6)} satisfy the given inequality.

[tex]6\ge\frac{1}{4}(4)+5[/tex]  ;[tex]\implies 6\ge6[/tex]: True

[tex]8\ge\frac{1}{4}(8)+5[/tex]  ;[tex]\implies 8\ge7[/tex]: True

[tex]6\ge\frac{1}{4}(-3)+5[/tex]  ;[tex]\implies 6\ge4.25[/tex]: True

Therefore, the correct answer is B.

Answer:

Approach: Difference of Squares Pattern

[tex]4 {x}^{2} - 25 = (2x - 5)(2x + 5)[/tex]

Step-by-step explanation:

The given binomial is:

[tex]4 {x}^{2} - 25[/tex]

We can rewrite to obtain:

[tex] {(2x)}^{2} - {5}^{2} [/tex]

This is a difference of two squares, so we will factor using difference of squares pattern.

Recall that:

[tex] {a}^{2} - {b}^{2} = (a + b)(a - )[/tex]

If we let

[tex]a = 2x[/tex]

and

[tex]b = 5[/tex]

Then we can factor the given binomial to obtain:

[tex] {2x}^2 - {5}^{2} = (2x - 5)(2x + 5)[/tex]

[tex] \therefore4 {x}^{2} - 25 = (2x - 5)(2x + 5)[/tex]

Answer: The answer is C. it would change orientation and slant down.

Step-by-step explanation:

The two lines are just mirror images of each other. In this case, the slope is just the opposite, causing it to be the way it is. The y intercept is the same which also causes the mirror imaging.

Answer:

It would change orientation and slant down.

Step-by-step explanation:

Answer:

A. 15

B. Scalene

Step-by-step explanation:

8x-10+6x+10x+10=360

24x=360

x=15

8x-10

8(15)-10=110

6x

6(15)=90

10x+10

10(15)+10=160

The triangle is scalene because none of the three sides are equal to each other.

Answer:

12 in²

Step-by-step explanation:

Since the figures are similar

linear ratio = a : b , then

area ratio = a² : b²

linear ratio of sides = 8 : 12 = 2 : 3

ratio of areas = 2² : 3² = 4 : 9

let the area of smaller figure be x then

[tex]\frac{4}{x}[/tex] = [tex]\frac{9}{27}[/tex] ( cross- multiply )

9x = 108 ( divide both sides by 9 )

x = 12

Area of smaller figure is 12 in²

Answer:

B. [tex]r_{x-axis}(x,y) \circ R_0,90\degree)[/tex]

Step-by-step explanation:

The vertices of triangle ABC have coordinates A(5,2) B(2,4) and C(2,1).

The mapping for reflection in the x-axis is

[tex](x,y)\to (x,-y)[/tex]

When we reflect triangle ABC in the x-axis, we obtain

A1(5,-2) B1(2,-4) and C1(2,-1).

The mapping for 90 degrees clockwise rotation about the origin is

[tex](x,y)\to (y,-x)[/tex]

When we rotate the resulting triangle through 90 degrees clockwise above the origin, we obtain;

A2(-2,-5) B2(-4,-2) and C2(-1,-2).

The vertices of triangle A''B''C'' also have coordinates A''(-2,-5) B''(-4,-2) and C''(-1,-2).

Hence the rule that describes the composition of transformation that maps ABC to A''B''C'' is

[tex]R_0,90\degree \circ r_{x-axis}(x,y)[/tex]

The correct choice is B.

Answer:

  an = 5·2^(n-1)

Step-by-step explanation:

The explicit formula for a geometric sequence with first term a1 and common ratio r is ...

  an = a1·r^(n-1)

Your sequence has a1=5 and r=2, so the explicit formula is ...

  an = 5·2^(n-1)

Final answer:

The explicit formula for the geometric sequence 5, 10, 20, 40, 80, 160,... is ,[tex]a_{n}=5(2^{n-1} )[/tex], where 'aₙ' represents the nth term in the sequence.

Explanation:

The student is asking for the explicit formula for a given geometric sequence. In mathematics, a geometric sequence is one where any term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. Looking at the sequence provided (5, 10, 20, 40, 80, 160,...), we can observe that each term is twice the previous term, thus the common ratio is 2.

To find the explicit formula of a geometric sequence, we use the formula ,[tex]a_{n}=a_{1} (r^{n-1} )[/tex], where an is the nth term, a₁ is the first term in the sequence, r is the common ratio, and n is the term number. For this sequence, a₁ is 5, and r is 2.

The explicit formula for the given geometric sequence is therefore [tex]a_{n}=5(2^{n-1} )[/tex].

Answer:

B

Step-by-step explanation:

It is less than 50% and more tha 10%

It’s B I just took the test or quiz

Answer:

176 cubic feet

Step-by-step explanation:

volume of each box is given by area of base * height.

Volume of Box 1 = 16 * 3 = 48

Volume of Box 2 = 16 * 3 = 48

Volume of Box 3 = 16 * 5 = 80

Total Volume = 48 + 48 + 80 = 176 ft^3

To calculate the total volume of the three boxes, we sum the individual volumes of two boxes at 48 cubic feet each and one box at 80 cubic feet, resulting in a total of 176 cubic feet.

To find the total volume of the three boxes shipped on a truck, we need to calculate the volume of each box and then sum the volumes. The volume of a rectangular prism (which is the shape of the boxes) is calculated by the formula Volume = length × width × height. In this case, the boxes have a common base of 16 square feet.

The two boxes with a height of 3 feet each have a volume of 16 square feet × 3 feet = 48 cubic feet per box. For both, this gives us a total of 48 cubic feet × 2 = 96 cubic feet.

The third box with a height of 5 feet has a volume of 16 square feet × 5 feet = 80 cubic feet.

Adding the volumes of all three boxes together, the total volume is 96 cubic feet + 80 cubic feet = 176 cubic feet. This is the total volume of the three boxes combined.

Answer:

option D

Step-by-step explanation:

So we know that [tex]\sqrt{a} \sqrt{b} = \sqrt{ab}[/tex]

Applying this to our function, we have that:

[tex]\sqrt{(x-5)} \sqrt{(x+2)} = \sqrt{(x+2)(x-5)}[/tex]

We know that the argument of a square root should always BE POSITIVE.

So we need to evaluate in which points the expression (x+2)(x-5) is positive.

So we know that (x+2) is possitive when x>-2 and negative when x<-2.

Also we know that (x-5) is possitive when x>5, if x<5 then x is negative.

Then we have:

if x<-2, then:

(x-5) is negative

(x+2) is negative

Then (x+2)(x-5) is positive.

If x>5 then:

(x-5) is positive

(x+2) is positive.

Then (x+2)(x-5) is positive.

If -2<x<5 then:

(x-5) is negative

(x+2) is positive

Then (x+2)(x-5) is negative, so it's undefined.

So the function is defined for x>5 and x<-2

So the correct option is option D.

Answer:

To quickly solve this problem, we can use a graphing tool or a calculator to plot each equation.

Please see the attached image below, to find more information about the graph

s

The equations are:

1) y = tan (x)

2) y = cot (x)

3) y = -tan (x)

4) y = -cot (x)

Looking at the graphs, we can see which corresponds to each equation

1) y = tan (x)

Graph C

2) y = cot (x)

Graph A

3) y = -tan (x)

Graph B

4) y = -cot (x)

Graph D

Answer:

A) 1C, 2A, 3B, 4D

Step-by-step explanation:

We can graph each of the functions in your preferred grapher, the we compare each graph with the corresponding example. (attached images)

By doing this we can see that graph C is tanx, graph A is cotx, graph B is -tanx, graph D is -cotx

[tex]175 - 14[/tex]

I think that will help you I'm not sure you just have to try to resolve if you get ranked if you get it wrong just send me a message cuz I am a fifth grade teacher

Answer:

Step-by-step explanation:

since the order of choosing out students to hand out papers doesnt matter because the paper handed is just the same and that the purpose or task of each student is just the same, combination should be used. Otherwise, when the order is based on ranking of students for example, permutation is used. answer is true

Answer:

8x^2 - 14xy + 3y^2

Step-by-step explanation:

You need to find the product of (2x - 3y )(4x - y ). To solve this, we're going to be using the distributive distribution as follows:

(2x - 3y )(4x - y ) = 8x^2 - 2xy - 12xy + 3y^2

Combining like-terms:

(2x - 3y )(4x - y ) = 8x^2 - 14xy + 3y^2

Therefore, the result is: 8x^2 - 14xy + 3y^2

Answer:

8x^2 - 14xy - 3y^2

the other explains it, it just forgets to factor in a negitive

Answer:

limaçon (no inner loop)

Step-by-step explanation:

The equation of a limaçon is written generically as ...

r = b + a·cos(θ)

For b=2 and a=1, this would put the flat side on the left. Rotating it 180° can be accomplished by adding or subtracting π from θ. Then the equation can be ...

r = 2 + cos(θ-π) . . . . or ...

r = 2 - cos(θ)

Answer:

t= 3 years

Step-by-step explanation:

So far we have 2 useful relationships[tex]P_{t}= P_{o} e^{r*t}[/tex] and [tex]P_{t}=2P_{0}[/tex]

Now, clearing t

[tex]P_{t}= P_{o} e^{r*t} \\\frac{ P_{t}}{P_{o}} = e^{r*t}\\\frac{2 P_{0}}{P_{o}} = e^{r*t}\\2 = e^{r*t}[/tex]

I apply logarithm

[tex]log(2)=r*t\\ t=\frac{log(2)}{r}\\t=\frac{0.3}{0.1} \\ t=3[/tex] years

Done

Answer:

Option C is correct

Step-by-step explanation:

First we will convert the mixed fractions into fraction form.

5/4, 5/8, 3/4, 1/2, 3/2, 7/4

Now, to find the range the formula used is:

Range= Maximum Value - Minimum Value

To find the maximum value and minimum value in fractions, we need to make the denominator of each fraction same. Since the highest value of denominator is 8, so other fractions denominator should be 8.

10/8, 5/8, 6/8,4/8,12/8,14/8

Maximum Value = 14/8 (having highest numerator)

Minimum value = 4/8 (having lowest numerator)

Range of data set = Maximum Value - Minimum value

                             = 14/8 - 4/8

                             = 10/8 = 5/4

                             = 1 1/4

so, option C is correct.

Answer:

The answer is C. 1 1/4

Step-by-step explanation:

To get the range of a set of data, all you need to do is get the difference between the highest value and the lowest value. Since you have different fractions, you first need to determine which is the highest and the lowest.

Since your given are not similar fractions (they do not have the same denominator), you need to make them similar first.

What I did first is make all mixed fractions into improper to make it easier for me to make them similar.

[tex]1\dfrac{1}{4},\dfrac{5}{8},\dfrac{3}{4},\dfrac{1}{2},1\dfrac{1}{2},1\dfrac{3}{4}\\\\\\\dfrac{5}{4},\dfrac{5}{8},\dfrac{3}{4},\dfrac{1}{2},\dfrac{3}{2},\dfrac{7}{4}[/tex]

The LCD of all fractions is 8 so I will need to get the proper proportion that will make all of them have 8 as the denominator.

[tex]\dfrac{5}{4}\times\dfrac{2}{2}[/tex] , [tex]\dfrac{5}{8}\times\dfrac{1}{1}[/tex] , [tex]\dfrac{3}{4}\times\dfrac{2}{2}[/tex] , [tex]\dfrac{1}{2}\times\dfrac{4}{4}[/tex] , [tex]\dfrac{3}{2}\times\dfrac{4}{4}[/tex] , [tex]\dfrac{7}{4}\times\dfrac{2}{2}[/tex]

So now we have the new set of fractions to compare:

[tex]\dfrac{10}{8}, \dfrac{5}{8},\dfrac{6}{8}, \dfrac{4}{8}, \dfrac{12}{8},\dfrac{14}{8}[/tex]

Looking at the set, all you have to do is get the fraction with the highest numerator (which will be our highest value) and get the fraction with the lowest numerator (which is the lowest value), then get the difference.

[tex]\dfrac{14}{8}-\dfrac{4}{8} = \dfrac{10}{8}[/tex] or [tex]1\dfrac{2}{8}[/tex]

Simplify it and you will get:

[tex]1\dfrac{1}{4}[/tex]

Final answer:

The recursive rule for the given sequence is [tex]a_n = a_{n-1} - 13.8[/tex],

where a₁ = -7.4 and n > 1.

Explanation:

You are looking for the recursive rule for the sequence -7.4, -21.2, -35, -48.8, -62.6, and so on. To find this, we observe how the sequence progresses from one term to the next. The pattern here is that each term decreases by the same amount when compared to the previous term. By calculating the difference between successive terms, we can identify the common difference.

For instance, the second term (-21.2) minus the first term (-7.4) equals the third term (-35) minus the second term (-21.2), and this difference equals -13.8. Hence, each term is -13.8 less than the term before it.

To express this as a recursive formula, we start by stipulating the first term:

Then, we provide the recursive rule that relates each term to the one before it:

Using this recursive formula, given any term in the sequence, we can find the next term by subtracting 13.8 from the given term.

Answer:

c

Step-by-step explanation:

Answer:

b.120in3

Step-by-step explanation:

you just take the high width and length and times together

4*5*6=120