Applications of Basic Equations:
97.) An isosceles triangle has two congruent sides that measure 3x + 8 in. and - 9x-5 over 2 in. respectively. The third sidle
28 over 2y-1
measures in. If the perimeter of the triangle is 72 in., find the length of all three sides. Find the value of y. Hint:
solve for x first.

Answer:

-6, 7

Step-by-step explanation:

Answer: B(-6,7)

Step-by-step explanation:

M(-5, 1)      A(-4,-5)       B(x,y)=?

[tex]\displaystyle\\M_x=\frac{x_A+x_B}{2} \\\\-5=\frac{-4+x_B}{2}\\\\[/tex]

Multiply both parts of the equation by 2:

[tex]-5(2)=-4+x_B\\\\-10=-4+x_B\\\\-10+4=-4+x_B+4\\\\-6=x_B\\\\Thus,\ x_B=-6[/tex]

[tex]\displaystyle\\\\\\1=\frac{-5+y_B}{2} \\\\[/tex]

Multiply both parts of the equation by 2:

[tex]1(2)=-5+y_B\\\\2=-5+y_B\\\\2+5=-5+y_B+5\\\\7=y_B\\\\Thus,\ y_B=7[/tex]

The final solution is:

(1) Tu=<7, 1, 13>, Tv=<0, -1, 1> and n(u,v)=<14, -7, -7>.

(2) the area of S=Φ(D): Area(S) = 514.39

(3) ∬sf(x,y,z)ds = 190√(294) = 3257.82

What is the double integral?

A multiple integral is a definite integral of a function of several real variables, for instance, f or f.

1. Φ(u,v) gives the parametrization with its x, y, and z components. So, x=7u+4, y=u-v, and z=13u+v.

We can use this to get the tangent vectors by taking the respective partial derivatives.

Tu=<7, 1, 13> by taking the u derivatives of the components, and Tv=<0, -1, 1> by taking the v derivatives of the components.

2. Taking the cross product of Tu and Tv will give the normal vector n(u,v), which is <14, -7, -7>.

3. To find Area(S), you have to multiply the area of D by the magnitude of the normal vector from the previous step. D is the region defined by 0<=u<=5 and 0<=v<=6, so the area of D is 5*6=30.

Multiply this by the magnitude of the normal vector to find that Area(S)=30√(294)

Area(S) = 514.39

4. To integrate, we first must put the initial function in terms of the parameters we found in the first step.

Replace y with u-v and z with 13u+v. Next, multiply this integrand by the magnitude of the normal vector (√294) and apply the given bounds for u (0<=u<=5) and v (0<=v<=6).

From here the problem can be integrated like any other double integral, the final answer being 190√(294).

Hence, The final solution is:

(1) Tu=<7, 1, 13>, Tv=<0, -1, 1> and n(u,v)=<14, -7, -7>.

(2) the area of S=Φ(D): Area(S) = 514.39

(3) ∬sf(x,y,z)ds = 190√(294) = 3257.82

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Complete Question:

Show that Φ(u,v)=(7u+4,u−v,13u+v) parametrizes the plane 2x−y−z=8. Then: (a) Calculate Tu , Tv, and n(u,v). (b) Find the area of S=Φ(D), where D=(u,v):0≤u≤5,0≤v≤6. (c) Express f(x,y,z)=yz in terms of u and v and evaluate ∬sf(x,y,z)ds(a) Tu= , Tv= , n(u,v)=_______________(b) Area(S)=_______(c) ∬sf(x,y,z)ds=___________

The variance of the number of heads which come up when a fair coin is flipped 9 times is 2.25.

In this case, number of trials n = 9

Note that X = X₁ + X₂ + X₃ + X₄ +X₅ + X₆ + X₇ + X₈ + X₉

Xₐ = 0 if the flip #i is tails, and

Xₐ = 1 if the flip #i is heads.

Since the Xi are independent, then we have:

Var (X) = Var (X₁ + … + X₉)

= Var (X₁) + Var (X₂) + … + Var (X₉)

Then,

Var (Xₐ) = E(Xₐ^2) – E(Xₐ)^2

= 1/2 – 1/4

= 1/4

So, the variance is:

Var (X) = Var (X₁) + Var (X₂) + … + Var (X₉)

= 9 * (1/4)

= 9/4

= 2.25

Hence, the variance of the number of heads which come up when a fair coin is flipped 9 times is 2.25.

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Answer:

II) it has all sides of equal length

To calculate how much Kaitlyn will have after four years with continuous compounding, we can use the following formula:

A = P * e^(r*t)

where A is the final amount, P is the initial principal (in this case, $3,000), r is the annual interest rate (4.08%), and t is the number of years (4). Plugging these values into the formula, we get:

A = $3,000 * e^(0.0408*4)

= $3,000 * e^0.1632

= $3,000 * 1.1759

= $3,527.77

Therefore, Kaitlyn will have $3,527.77 after four years with continuous compounding, which is close to [A] $3,532.18.

To determine the length of the picture, we can set up the following equation:

length = 4 * width

Substituting in the given value for the width, we get:

length = 4 * 24 inches

Solving this equation gives us a length of 96 inches. Therefore, the length of the picture is 96 inches.

The surface area of a sphere is given by the formula 4πr^2, where r is the radius of the sphere. This means that the surface area of the balloon is increasing at the rate of 8πr cm^2/sec.

The volume of a sphere is given by the formula 4/3πr^3, where r is the radius of the sphere. If the radius of the balloon is 6 cm, then the volume of the balloon is (4/3)π(6^3) = (4/3)π216 = 288π cm^3.

To find the rate at which the volume of the balloon is increasing, we need to take the derivative of the formula for the volume of a sphere with respect to time. This gives us (4/3)π(3r^2)(dr/dt).

Since the radius of the balloon is 6 cm and the surface area of the balloon is increasing at the rate of 8πr cm^2/sec, the rate at which the radius of the balloon is increasing is (8πr/4πr^2) = 2/r = 2/6 = 1/3 cm/sec.

Plugging this into the formula for the rate of change of the volume of the sphere, we get

Therefore, the volume of the balloon is increasing at a rate of 144π cm^3/sec when the radius of the balloon is 6 cm.

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Complete Question:

The surface area of a spherical balloon is increasing at the rate of 2 cm 2/sec. At what rate is the volume of the balloon is increasing when the radius of the balloon is 6 cm?

Answer:

6

Step-by-step explanation:

        6 is a factor of X , Y and Z.

    So, 6 will be a common factor of X + Y + Z.

Answer: 1. 108.     2. 1         mark brainlist if helpful

Step-by-step explanation: hope it helps

98 plus 10.   108 plus one 109.  

g(x) intercepts the x-axis at x = 3, then the correct option is (3, 0)

For a function y = f(x), we define the x-intercept as the value of x for which:

f(x)=  0

The x-intercepts are also called zeros or roots of the function, and are the points where the graph of the function intercepts the x-axis.

In this case, the function is g(x) = -3x + 9, so here we need write and solve the equation:

g(x) = 0

-3x + 9 = 0

9 = 3x

9/3 = x

3 = x

This means that the x-intercept is x = 3, or (3, 0) written in point form, when we evaluate g(x) in x = 3 we get:

g(3) = -3*3 + 9

g(3) = -9 + 9

g(3) = 0

As expected.

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A level curve of the function f(x, y) is a curve that satisfies the equation f(x, y) = c, where c is a constant.

The diagram is attached at the end of the solution.

A contour map, which is also called a topographic map, is a representation of a three-dimensional feature using contour lines on a plane surface.

The map shows a bird's-eye view and allows people to visualize the hills, valleys, and slopes that are being mapped.

People can see the hills, valleys, and slopes that are being mapped thanks to the map's bird's-eye perspective.

The title, scale, contour interval, legend, and whether latitude and longitude or Universal Transverse Mercator (UTM) coordinates are used are typically included.

Numerous activities, such as camping, urban planning, meteorology, and geologic investigations, can benefit from using this kind of map.

Consider the surface [tex]$f(x, y)=\ln \left(x^2+4 y^2\right)$[/tex].

The level curves for the surface are given by the equation, [tex]$k=\ln \left(x^2+4 y^2\right)$[/tex], where [tex]$k \in \mathbb{R}$[/tex]

Take k = . . . , -3, -2, -1, 0, 1, 2, 3, . . .

Then, draw the different level curves of the surface.

The diagram is attached at the end of the solution.

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b. The correlation coefficient for this data-set is given as follows: r = -0.9877.

c. The correlation between the variables x and y is strong negative.

The correlation coefficient is an index between -1 and 1 that measures the relationship between two variables, as follows:

A data-set is composed by a set of points, and these points are inserted into a correlation coefficient calculator to obtain the coefficient.

From the table described in this problem, the points are given as follows:

(1, 9), (2, 7), (3,6), (4, 1), (7, -2), (8, -5) and (10, -8).

Inserting these points into a calculator, the coefficient is given as follows:

r = -0.9877.

Hence it is a negative and strong relationship, as the absolute value of r is of 0.9877 > 0.6.

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(i) Yes, it is even and divisible by 3 as 1 + 0 + 2 = 3

(ii) No, it does not end in 5 or 0

What are Divisibility Rules?

Divisibility criteria can be used to determine whether or not to diminish a fraction. The rules are based on patterns seen while listing the multiples of any integer.

Solution:

(i) 102 is divisible by 6 - TRUE

It is because 102 is even number thus, it is divisible by 2 and since the sum of digits 1 + 0 + 2 is 3 therefore, it is divisible by 3

- Yes, it is even and divisible by 3 as 1 + 0 + 2 = 3

(ii) 102 is divisible by 5 - FALSE

It is because any number to be divisible by 5 it should always end with 5 or 0 at ones place.

- No, it does not end in 5 or 0

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Answer:

4 ( 2q + 5 )

Step-by-step explanation:

8q + 20

HCF is 4

= 4 ( 2q + 5 )

pls rate as brainliest

Answer:

Step-by-step explanation:

ok so standard form is

Ax+By=C

and its actually really simple

when you want to find the y-intercept

set x to 0

it will look something like

A(0)+By=C

By=C

y=C/B

It makes more sense with actual numbers

so

3x+2y=4

3(0)+2y=4

0+2y=4

2y=4

y=2

therefore the y intercept would be 2

for that equation

but you can do the same for x-intercept

just set y to 0

hope this helped :)

The relationship represented is linear. Where f(0) = 2 and f(n) = f(n-1)+4.

A linear function in mathematics is one that has either one or two variables and no exponents. It is a function with a straight line as its graph. The graph of a linear function is a straight line. The following is the form of a linear function. a + bx = y = f(x). One independent variable and one dependent variable make up a linear function. X and Y are the independent and dependent variables, respectively. There are only one or two variables in a linear equation. No variable can be multiplied by a number larger than one or used as the denominator of a fraction in a linear equation.

Given,

Functions,

f(0) = 2

f(n) = f(n-1) + 4

Equating 0

f(0) = f(0-1) + 4

f(0) = 4

The relationship represented is linear. Where f(0) = 2 and f(n) = f(n-1)+4.

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The probability that to total time to wash and condition a customer's hair will take longer more than 5 minutes is  0.63341

In this question we have been given that at supercuts, washing a customer's hair takes an average of 3. 5 minutes with a standard deviation

of 1. 2 minutes. Applying conditioner takes an average of 2. 1 minutes with a standard deviation of. 75 minutes.

μ1 = 3.5, μ2 = 2.1

σ1 = 1.2, σ2 = 0.75

The mean of the sum of the two random variables would be,

μ = μ1 + μ2

μ = 3.5 + 2.1

μ = 5.6

The Standard Deviation of the Sum of Two Independent Random Variables would be,

σ = √(σ1)² + (σ2)²

σ = √(1.2)² + (0.75)²

σ = 1.76

The probability that to total time to wash and condition a customer's hair will take longer more than 5 minutes.

i.e., P(x > 5)

First we find the z-score.

z = X - μ/σ

z = 5 - 5.6/1.76

z = -0.34091

P-value from Z-Table:

P(x<5) = 0.36659

So, P(x>5) = 1 - P(x<5)

                 = 0.63341

Therefore, the required probability is  0.63341

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The area enclosed by the curve x = t^2 − 3t, y = √t and the y-axis is 2.08 square units.

We have been given parametric equations x = t^2 − 3t, y = √t

We need to find the area enclosed by the curve x = t^2 − 3t, y = √t and the y-axis.

Consider x = 0

So, t^2 − 3t =  0

t(t - 3) = 0

t = 0   or  t = 3

Let f(t) =  t^2 − 3t and g(t) = t

Differentiate the curve f(t) with respect to t.

f'(t) = 2t - 3

NWe know that the formula to find the area under the curve.

A = ∫[a to b] g(t)f'(t) dt

here, a = 0 and b = 3

so, A = ∫[0 to 3] √t (2t - 3) dt

A = ∫[0 to 3] (2t√t - 3√t) dt

A = ∫[0 to 3] (2t^(3/2) - 3t^(1/2)) dt

A = [4/5 t^(5/2) - 2 t^(3/2)]_[t = 0, t = 3]

A = 4/5 3^(5/2) - 2 3^(3/2) - 0 + 0

A = 4/5 3^(5/2) - 2 3^(3/2)

A = 6√3 /5

A = 2.08

Therefore,  the area of the curve is 2.08 square units.

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The slope of the rapids is -0.067

Kannanaski Rapids drops  62 ft vertically

The horizontal distance of 920 ft.

tan θ is a commonly used trigonometric function along with other 5 functions. tan θ is also called as law of tangent. The tangent formula for a right-angled triangle can be defined as the ratio of the opposite side of a triangle to the adjacent side. It can also be represented as a ratio of the sine of the angle to the cosine of the angle.

Slope of Rapids= tanθ= [tex]\frac{y}{x}=\frac{-62}{920} \\[/tex]

=-0.067

Therefore, the slope of rapids is -0.067.

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Answer:

y=1/2x+1

Step-by-step explanation:

I think so I'm not an expert

Answer:

y = - [tex]\frac{1}{2}[/tex] x - 1

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

calculate m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (2, - 2 ) and (x₂, y₂ ) = (- 4, 1 )

m = [tex]\frac{1-(-2)}{-4-2}[/tex] = [tex]\frac{1+2}{-6}[/tex] = [tex]\frac{3}{-6}[/tex] = - [tex]\frac{1}{2}[/tex] , then

y = - [tex]\frac{1}{2}[/tex] x + c ← is the partial equation

to find c substitute either of the 2 points into the partial equation

using (2, - 2 ) , then

- 2 = - 1 + c ⇒ c = - 2 + 1 = - 1

y = - [tex]\frac{1}{2}[/tex] x - 1 ← equation of line

The smaller number is 25 and the larger number is 30.

Given that, the sum of two numbers is 55.

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

Let the smaller number be x.

The larger number is 5 more than the smaller number.

Then, the larger number be x+5

The sum of two numbers is 55.

x+x+5=55

⇒ 2x+5=55

The solution of an equation is the set of all values that, when substituted for unknowns, make an equation true.

Now, 2x+5=55

⇒ 2x=50

⇒ x=25

So, x+5=30

Therefore, the smaller number is 25 and the larger number is 30.

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Answer:

No it is not possible, you would have to do 74.25 divided by 3

Step-by-step explanation:

3 divided by 74.25 = 0.0404040404

but

74.25 divided by 3 = 24.75

Answer:

£22.33

Step-by-step explanation:

calculate the cost of an album and a single track using simultaneous equations.

let a represent an album and s a single track , then

3a + 4s = 18.13 → (1)

2a + 5s = 13.93 → (2)

multiplying (1) by - 2 and (2) by 3 and adding will eliminate a

- 6a - 8s = - 36.26 → (3)

6a + 15s = 41.79 → (4)

add (3) and (4) term by term to eliminate a

0 + 7s = 5.53

7s = 5.53 ( divide both sides by 7 )

s = 0.79

substitute s = 0.79 into either of the 2 equations and solve for a

substituting into (1)

3a + 4(0.79) = 18.13

3a + 3.16 = 18.13 ( subtract 3.16 from both sides )

3a = 14.97 ( divide both sides by 3 )

a = 4.99

that is album costs £4.99 and single track £0.79

then cost to Charlotte is

cost = (4 × £4.99) + (3 × £0.79) = £19.96 + £2.37 = £22.33

The cost of 4 albums and 3 single tracks is £22.33

Let the cost of 1 album = £x

and let the cost of 1 single track = £y

Now, the cost of 3 albums and 4 single tracks = £3*x + £4*y

and cost of 2 albums and 5 single tracks= £2*x + £5*y

Given that,

£3*x + £4*y = £18.13  -------->(1)

£2*x + £5*y = £13.93 -------->(2)

Multiply (1) equation by 2 and (2) equation by 3 then solve for x and y.

On applying the substitution method we have,

x = £4.99 and y = £0.79

Now, we have to calculate the value of 4 albums and 3 single tracks.

It can be written as 4*x +3*y

£4*4.99 + £3*0.79 = £22.33

So, the cost of 4 albums and 3 single tracks is £22.33

This question is based on a linear equation in two variables. Here, we used two equations based on two variables and then solved them with the help of the substitution method.

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According to the Kepler's third law, the squares of the period of a planet's orbit is proportional to the cube of the semimajor axis.

According to Kepler's third law,

     t2 = a3

Kepler's third law compares the orbital period and radius of orbit of a planet to those of other planets. This comparison between period and radius being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets. It provides an accurate description of the period and distance for a planet's orbits about the sun. The square of the period of a planet's orbit is proportional to the cube of its semimajor axis.  Kepler's Third Law depends on the total mass of the two bodies involved.

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Answer:

square and cube is the answers

Step-by-step explanation:

An expression to represent the total penalty fees after 3 days late is given by T = 25d.

In Mathematics, a linear function is sometimes referred to as an expression or the slope-intercept form of a straight line and it can be used to model (represent) the total penalty fees after 3 days late;

T = md + b

Where:

Therefore, the required linear function that represents the total penalty fees after 3 days late is given by this mathematical expression;

T = 25d

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Answer:

C. y=4/3x-1

Step-by-step explanation:

I plotted all equations on Desmos. A, B and D all had a solution with the equation of line a, they both had an interception so they were wrong. C. is the only that does not have an interception with line a. They didn't intercept and were parallel meaning that they will never touch or won't have a solution.

Answer:

145°

Step-by-step explanation:

Angle x and the 35-deg angle are a liner pair, so they are supplementary.

The sum of the measures of two supplementary angles is 180°.

x + 35° = 180°

x = 145°

From the given options, option B is the only statement that is not true is ∠C ≅ ∠E.

Triangle congruence theorem or triangle congruence criteria help in proving if a triangle is congruent or not. The word congruent means exactly equal in shape and size no matter if we turn it, flip it or rotate it.

If Δ ABC is congruent to Δ DEF, then its corresponding parts must be congruent.

segment AB ≅ segment DE

segment BC ≅ segment EF

segment AC ≅ segment DF

∠A ≅ ∠D

∠B ≅ ∠E

∠C ≅ ∠F

Therefore, from the given options, option B is the only statement that is not true is ∠C ≅ ∠E.

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"Your question is incomplete, probably the complete question/missing part is:"

If triangle ABC is congruent to triangle DEF, which statement is not true?

A) Segment AB ≅ segment DE

B) ∠C ≅ ∠E

C) Segment BC ≅ segment EF

D) ∠A ≅ ∠D