The age of Mark’s dad is 4 less than twice Mark’s age. The sum of their ages is 95. Find their ages.

The surface area increases when the radius of the sphere is 160π cm^2/s

Given that dt/dr =10 cm/s and r=15 cm

Find out dt/ds=?

We know that the surface area of sphere S=4πr^2

dt/ds=4π(2r)

dt/dr=8π(2)(10)

=160π cm^2/s.

Let r, V, and S be respectively the radius, volume, and surface area of the spherical balloon at time t.

∴V=4/3 πr^3 and S=4πr^2

Rate of change of surface area with respect to t

⇒ dt/dS=2

⇒ d/dt=(4πr^2)=2

⇒8πr dt/dr =24

⇒ dr/dt=1/4πr.... (i)

Rate of change of volume of the balloon with respect to t

=dV/dt

=d/dt(4/3πr^3)

= 4/3π⋅3r^2dr/dt

​=4πr ^2⋅ dr/dt

=4πr^2.1/4πr ....[From (i)]

r=10

Hence, the volume of the spherical balloon is increasing at the rate of 10cm^3/sec.

To know more about the Area of the sphere:

https://brainly.com/question/96130

#SPJ4

given that v1,v2 ,v3,v4 is linearly independent vector so in order for a set of vectors to be linearly independent it must satisfy the below conditon:

   [tex]x_1v1+x_2v2+x_3v3+x_4v4[/tex] =0  --(i)

  where [tex]x_1, x_2,x_3 ,x_4[/tex] is some real numbers.

a) so in order to show that v1,v2+v4,v3 is linearly independent we have to

check if equations formed by these vectors can be transformed into some equation of equation (i) form.

 so we have :

   [tex]x_1v1+x_2(v2+v4) +x_3v3[/tex]   ---(ii)

    where [tex]x_1, x_2,x_3[/tex] is some real numbers.

  after evaluating equation (ii) we have:

     [tex]x_1v1+x_2v2+x_3v3+x_2v4[/tex]  

     so the above equation is now transformed in the form of equation (i) where [tex]x_2 = x_4[/tex]  

and hence [tex]x_1v1+x_2v2+x_3v3+x_2v4[/tex] =0  ----(iii)

equation (iii) shows that the set of given vectors are linearly independent.

b)  so in order to show that v1 +v2, v3, v4  is linearly independent we have to check if equations formed by these vectors can be transformed into some equation of equation (i) form.

 so we have this equation:

   [tex]x_1(v1+v2)+ x_2v3 + x_3v4[/tex]      where [tex]x_1, x_2,x_3[/tex] is some real numbers.

  after evaluating equation (iv) we have:

     [tex]x_1v1+x_1v2+x_2v3+x_3v4[/tex]  

     so the above equation is now transformed in the form of equation (i) where the cofficient of v2 is also [tex]x_1[/tex]  

and hence [tex]x_1v1+x_1v2+x_2v3+x_3v4[/tex] =0  ----(v)

equation (v) shows that the set of given vectors are linearly independent.

To know more about vector click on below link:

https://brainly.com/question/15709504#

#SPJ4

Complete question :

let v be a real vector space. suppose that v1, v2, v3, v4 are vectors in v which are linearly independent. show that the vectors a) v1, v2+v4, v3 and   b) v1 +v2, v3, v4 are also linearly independent.

The vectors v1, v1+v2, v1+v2+v3, v1+v2+v3+v4 are also linearly independent is hence proved.

let v be a real vector space.

The v1, v2, v3, v4 are vectors in v which are linearly independent.

V={v1, v2, v3, v4} then set V is called  linearly independent if [tex]\forall \alpha_i \in F \\[/tex].

∝1v1+∝2v2+∝3v3+..........+∝nvn=0

then, [tex]\forall \alpha_i =0 \\[/tex] i=1,2,3......n.

let ∝1,∝2,∝3,∝4 ε F

∝1v1+∝2(v1+v2)+∝3(v1+v2+v3)+∝4(v1+v2+v3+v4)=0

v1(∝1+∝2+∝3+∝4)+v2(∝2+∝3+∝4)+v3(∝3+∝4)+v4.∝4=0

Since  v1, v2, v3, v4 are linearly independent vectors.

So ∝1+∝2+∝3+∝4=0......(1)

∝2+∝3+∝4=0....(2)

∝3+∝4=0....(3)

∝4=0...(4)

Solving eq(1), (2), (3), (4)

we get ∝1=∝2=∝3=∝4=0

So according to definition of linearly independent vectors.

A set of vectors is called linearly independent if no vector in the set can be expressed as a linear combination of the other vectors in the set. If any of the vectors can be expressed as a linear combination of the others, then the set is said to be linearly dependent.

So v1, v1+v2, v1+v2+v3, v1+v2+v3+v4 are also linearly independent vectors hence proved.

For more such questions about linear independent vector

https://brainly.com/question/23878734

#SPJ4

Answer:

44 unit^2.

Step-by-step explanation:

Divide the figure into 2 trlangles, a rectangle and a trapezium

Area = 2 * 1/2 *4 * 3 + 2 ** + 1/2 * 8 * (1 + 3)

        = 12 + 16 + 16

         = 44.

The cost of admission for a group of 4 people is $160

From the question, we have the following parameters that can be used in our computation:

C = 40 p

Where p is the number of people

For a group of 4 people, we have

p = 4

Substitute p = 4 in the equation C = 40 p

So, we have the following representation

C(4) = 40 * 4

Evaluate the products

C(4) = 160

hence, the cost is $160

Read more about linear functions at

https://brainly.com/question/15602982

#SPJ1

Answer: 55000 grams

Step-by-step explanation:

9/10 is the possible outcome that has a number less than 3 or an A.

What is probability?

The proportion of possible outcomes to outcomes in an exhaustive set of equally likely alternatives that result in a certain event.

Here, we have

A spinner is divided into five equal sections labeled 1, 2, 3, 4, and 5. Another spinner is divided into two equal sections labeled A and B.

We have to find how many of the possible outcomes have a number less than 3 or an A.

Denote A the first event and B the second event:

P(A∪B) = P(A) + P(B)

A = Get a number less than 3

P(A) = 2/5

B = Get section A

P(B) = 1/2

Then,

P(A∪B) = 2/5 + 1/2

P(A∪B) = 9/10

Hence, 9/10 is the possible outcome that has a number less than 3 or an A.

To learn more about the probability from the given link

https://brainly.com/question/24756209

#SPJ1

The value of the decrease percentage of elephant weight is 15%.

A percentage is a value that indicates 100th part of any quantity. A percentage can be converted into a fraction or a decimal by dividing it by 100.

The original weight of elephant is 14000 pounds.

And, the drop in weight is 2100 pounds.

Now, the percent value of the drop in weight is calculated as follows

Percent drop = Drop in weight / Original weight × 100

⇒ 2100/14000 × 100

⇒ 15%

Hence, the percent of decrease in weight is given as 15%.

To know more about percentage click on,

brainly.com/question/29306119

#SPJ1

slope -20/-8

line y = -5/2x - 3/2

it is NOT proportional because it doesnt pass thru 0,0

The number of Jim's model car is 48

From the question, we have the following parameters that can be used in our computation:

Ratio of the number of model cars that Jim owns to the number of model cars Terrence owns is 8 : 6

This means that

Jim : Terrence = 8 : 6

Also from the question, we have

Terrence owns 36 models cars

This means that

Terrence = 36

Substitute the known values in the above equation, so, we have the following representation

Jim : 36 = 8 : 6

Multiply by 6

Jim : 36 = 48 : 36

So, we have

Jim = 48

Hence, the number of car is 48

Read more about ratio at

https://brainly.com/question/1781657

#SPJ1

The correct answer is m√5 + m√2 = 180°.

Parallel lines are those in a plane that are regularly spaced apart from one another. There is no crossing of parallel lines.

Given, k and p are parallel lines.

In an expression or equation, one value can be substituted for another and the result will still be the same. This is known as the substitution property of equality. If the values of x and y are the same, then x and y can be substituted for one another.

And we have,

m√1 + m√2 = 180°

And  m√5 =  m√1

So,  m√5 + m√2 = 180°

Therefore, in the proof, the missing term is  m√5 + m√2 = 180°.

To learn more about the parallel lines;

brainly.com/question/16701300

#SPJ1

The domain and range of the function are (-∞,∞) and (-∞,5). The greatest concentration will be 5.

The set of all values that a function can accept is known as its range, while the set of all values for which a function is defined is known as its domain.

The dependent variable's range is its domain, which is for the independent variable.

Part (A)

As per the given function C(t) = - 5t² + 10t

C'(t) = 0

-10t + 10 = 0

10t = 10

t = 1

C''(t) = -10 < 0

So, at t = 1 the function will be maximum.

C(1) = -5(1)² + 10 x 1 = 5

Thus, the range will be (-∞,5) and the domain will be (-∞,∞).

The range represents the amount of concentration while the domain is the time.

Part (B)

The graph of the function has been drawn below.

At t = 1 the C(t) is maximum as 5 thus it will be the maximum concentration.

Hence"The function's domain and range are  (-∞,∞) and (-∞,5), respectively. Five will have the highest concentration.".

For more details about the range and domain of the function,

brainly.com/question/28135761

#SPJ1

The inverse function of the function f(x) = log₂(3x + 4) − 5 is y = [2^(x + 5) - 4]/3

From the question, we have the following equation that can be used in our computation:

f(x) = log2(3x + 4) − 5.

Rewrite as

f(x) = log₂(3x + 4) − 5.

Express f(x) as y

This gives

y = log₂(3x + 4) − 5.

Swap the positions of x and y

So, we have the following representation

x = log₂(3y + 4) − 5.

Add 5 to both sides

log₂(3y + 4) = x + 5

So, we have

3y + 4 = 2^(x + 5)

Subtract 4 from both sides

3y = 2^(x + 5) - 4

Divide b y3

y = [2^(x + 5) - 4]/3

Hence, the inverse function is y = [2^(x + 5) - 4]/3

Read more about inverse function at

https://brainly.com/question/14796161

#SPJ1

With the decimal point in the proper place, dividing decimals is equivalent to dividing entire numbers.

To Learn more About dividing decimals refer To:

https://brainly.com/question/11786124

#SPJ1

The Alternative form of the ''(2x4-2x³ +7+ 8x²-x) / (x²+2)''

is 2[tex]x^{2}[/tex] - 2x + 4 + [tex]\frac{3x - 1}{x^{2} + 2}[/tex]

Find the attachment answer

Learn more about algebra equation refer to :

https://brainly.com/question/24875240

#SPJ1

Answer:

1. range = {-8,-6,0,8}

2. My graph is attached to this response as a photo.

Step-by-step explanation:

1. ⭐What is the "domain" of a function?

⭐What is the "range" of a function?

⭐What is the "set notation"?

Therefore, we have to find the corresponding y-values for the given x-values, and then write said y-values in set notation.

To find the corresponding y-values for the given x-values, substitute each given x-value into g(x) and solve for g(x).

✦ When x = -3

[tex]g(x) = 2x-2[/tex]

[tex]g(-3) = 2(-3) -2[/tex]

[tex]g(-3) = -6 -2[/tex]

[tex]g(x) = -8[/tex]

✧ y = -8

________________________________________________________

✦ When x = -2

[tex]g(-2) = 2(-2) -2[/tex]

[tex]g(-2) = -4-2[/tex]

[tex]g(-2) = -6[/tex]

✧ y = -6

________________________________________________________

✦ When x = 1

[tex]g(1) = 2(1) -2[/tex]

[tex]g(1) = 2-2[/tex]

[tex]g(1) = 0[/tex]

✧ y = 0

________________________________________________________

✦ When x = 5

[tex]g(5) = 2(5) -2[/tex]

[tex]g(5) = 10 -2[/tex]

[tex]g(5) = 8[/tex]

✧ y = 8

________________________________________________________

Next, we have to write the y-values in set notation. Make sure to write it from least to greatest.

range = {-8, -6, 0, 8}

Cos W as a fraction in the simplest terms is 2/3  

Triangle side length ratios are known as trigonometric ratios. In trigonometry, these ratios show how the ratio of a right triangle's sides to each angle.

From a Right-angled triangle with an angle θ

sin θ = Opposite side/Hypotenuse

cos θ = Adjacent side/Hypotenuse

tan θ = Opposite side /Adjacent side

Here we have

A triangle UVW

where VW = 10 units

And WU = 15 units

As we know

cos θ = Adjacent side / Hypotenuse  

cos W = VW/WU = 10/15

cos W = 2/3    [ divided by 5 ]  

Therefore,

Cos W as a fraction in the simplest terms is 2/3  

Learn more about Trigonometric ratios at

https://brainly.com/question/25122832  

#SPJ1

As a result, the singers can finish first, second, third, fourth, or fifth in 524,160 different ways.

Permutation, where order is important:

_nP_{r}=\dfrac{n!}{(n-r)!}

P r = (n−r)!n!

Combination of definitions (order is irrelevant):

_nC_{r}=\dfrac{n!}{r!(n-r)!}n C r equals r!(nr)!n!

with n!=n, cdot (n-1) ..., cdot 2..., cdot 1n!=n(n1)...2,1.

We want to know how many different ways we can choose five people from a group of 16 (ranked first, second, third, fourth, and fifth).

n=16 n=16 r=5 r=5 Because we are interested in the first, second, third, fourth, and fifth places, order is important (changing the first and second places results in different rankings and prizes for the people), so we need to use a "textbf permutation" permutation.

Analyze the permutation definition:

"Begin align*" means "16P_5" and "dfrac" 16! color#8895cetextDefinition combination &=dfrac16!11!&color#8895cetext Evaluate difference &=dfrac16cdot 15cdot 14cdot 13cdot 12cdot 11cdot 10cdot...cdot 111cdot 10...cdot 1

16!= 11!

16! = 11⋅10⋅...⋅1

16⋅15⋅14⋅13⋅12⋅11⋅10⋅...⋅1

=16⋅15⋅14⋅13⋅12

=524,160

To learn more about Permutation here

https://brainly.com/question/1216161

#SPJ4

Full Question = There are 16 finalists in a singing competition. The top five singers receive prizes. How many ways can the singers finish first through fifth?

A right Riemann sum has the formula  A=∑ni=1Δxf(xi) A = ∑ i = 1 n Δ x f ( x i ) where x is the width of each of the n rectangles and f(xi) is the height.

A specific method of approximating an integral with a finite sum in mathematics is known as a Riemann sum. Riemann is the name of a German mathematician who lived in the 19th century. Approximating the area of functions or lines in graphs, as well as the length of curves and other approximations, is a highly typical application. The picked points in each subinterval of the partition are all in the upper Riemann sum, given an interval partition. The interval is often separated into subintervals of equal size.

Riemann sum formula is A=∑f(xi)Δx A = ∑ f ( x i ) Δ x ,

where A is the curve's area under the evaluable interval,

f(xi) f ( x i ) is each rectangle's height (or the average of the two heights in the case of a trapezoid)

To know more about Riemann sum visit:
https://brainly.com/question/29012686

#SPJ4

The length of diameter of the base of the cone will be 112 cm.

A mathematical expression with an equals sign is referred to as an equation. Algebra is widely used in equations. When performing calculations but unsure of the precise amount, algebra is used.

A formula that expresses the connection between two expressions on each side of a sign. Typically, it has a single variable and an equal sign.

Equations are mathematical expressions with two algebras flanking the equal (=) sign on either side. This relationship is illustrated by the left and right expressions being equal to one another. Left-hand side equals right-hand side is a basic, straightforward equation.

By using Pythagoras's theorem,

106² = 90² + r²

simplifying the above equation, we get

r² = 106² - 90²

r = √3136

radius = 56 cm.

Then, the diameter will be,

Diameter = 2 × radius

substitute the values in the above equation,

D = 2(56)

D = 112 cm.

Therefore, the correct answer is option b. 112 cm.

To learn more about Pythagoras's theorem refer to:

https://brainly.com/question/343682

#SPJ1

The slope of a line that is perpendicular to the x-axis is undefined, so we cannot use the standard form of the equation of a line, y = mx + b, where m is the slope of the line. Instead, we can use the point-slope form of the equation of a line, which is given by y - y1 = m(x - x1). In this case, m is undefined, so we can simplify the equation to y = x1 - y1, where x1 and y1 are the coordinates of the point that the line passes through. In our case, this gives us the following equation:

y = 6 - (-2)

 = 8

Therefore, the equation of the line that is perpendicular to the x-axis and passes through the point (6,-2) is y = 8.

Answer:

x = 44

Step-by-step explanation:

Given that line m is parallel to line n, the angles that we see are same-side exterior angles, meaning that these two angles add up to 180°.

Set up the equation: [tex]112 + (2x - 20) = 180[/tex] and solve for x.

. . . . . . . . . . . . 112 - 112 + 2x - 20 = 180 - 112

. . . . . . . . . . . . 2x - 20 + 20 = 68 +20

. . . . . . . . . . . . 2x /2 = 88 /2

. . . . . . . . . . . .  ∴ x = 44

Answer:

A.) y=6x+2

Step-by-step explanation:

I got this answer by first finding the slope. To find the slope you use the slope formula which is y2-y1 over x2-x1. When I set up the equation I got -10-26 over -2-4 and I got -36/-6 which is 6. So your slope is 6. So then you would mark out answer c and d because their slope is negative and the slope I got is positive. To find the y-int I put in two points online and I put in the slope. I did this twice with four different points just to make sure that it was right and then it gave me that the y-int is 2. So then you would mark out b because the y-int is not negative but positive. (Hope this helps in some way...if not then I'm sorry..)

The value of the expression f(x) + g(x) - h(x) will be 2x² + 7x + 2. Then the correct option is B.

A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.

The functions are given below.

f(x) = 3x² + 5

g(x) = 4x - 2

h(x) = x² - 3x + 1

The expression is given as,

⇒ f(x) + g(x) - h(x)

⇒ (3x² + 5) + (4x - 2) - (x² - 3x + 1)

⇒ 3x² + 5 + 4x - 2 - x² + 3x - 1

⇒ 2x² + 7x + 2

The value of the expression f(x) + g(x) - h(x) will be 2x² + 7x + 2. Then the correct option is B.

More about the function link is given below.

https://brainly.com/question/5245372

#SPJ1

The length of diameter of the base of the cone will be 112 cm. Hence, option b is correct.

Given data :

As per the given information in the question,

Use the equation of Pythagoras's theorem,

106² = 90² + r²

r² = 106² - 90²

r = √3136

r = 56 cm.

Then, the diameter will be,

D = 2r

D = 2(56)

D = 112 cm.

Learn more about equation of Pythagoras's theorem refer to :

https://brainly.com/question/24408180

#SPJ1

324 Miles

If you divide 189 by 7, you get 27. If you then multiply 27(miles per hour) by 12(hours) you get 324.

To find the distance that Jacob's truck skids before coming to a stop, we can use the formula for the distance traveled while braking, which is:

distance = initial velocity^2 / (2 * coefficient of friction * acceleration due to gravity)

In this case, the initial velocity is 9.5 m/s, the coefficient of friction is 0.847, and the acceleration due to gravity is 9.8 m/s^2. Plugging these values into the formula, we get:

distance = (9.5 m/s)^2 / (2 * 0.847 * 9.8 m/s^2)

Solving for distance, we get:

distance = 8.76 meters

Therefore, Jacob's truck will skid a distance of 8.76 meters before coming to a stop.

The values are satisfied by

X  -5    -4    -3    -2    -1      

Y -5     -3     -1      1     3

Given data :

Y= 2X +5

let us put the value for X

X= 0

Y = 5

and, X=1

Y = 2 + 5 = 7

and, X= 2

Y= 4 + 5 = 9

and, X=3

Y = 6 + 5 = 11

and, X= 5

Y = 10 + 5= 15

and, X= -1

Y= -2 + 5 = 3

Hence, these values are satisfied by

x      y

-5    -5

-4    -3

-3    -1

-2     1

-1      3

Learn more about graph a function refer to :

https://brainly.com/question/24335034

#SPJ1