Jason has $90 to spend. He wants to purchase a bag for $35,one eraser for $10, and three pencils. Each of the pencils costs the same price. This will use up all of Jason's money. Determine the price of a pencil.​

Answer:

Step-by-step explanation:

The height of the trapezoid is 9 yards. Therefore, the correct answer is option b. 9 yd.

To find the height of the trapezoid with the given area and base lengths, we will use the formula for the area of a trapezoid:

Area = (1/2) * (base1 + base2) * height

Here, the area is given as 126 square yards, base1 is 17 yards, and base2 is 11 yards. We need to find the height.

1. Substitute the given values into the formula:

126 = (1/2) * (17 + 11) * height

2. Simplify the equation:

126 = (1/2) * 28 * height

3. To isolate the height, divide both sides by (1/2) * 28:

height = 126 / ((1/2) * 28)

4. Calculate the result:

height = 126 / 14

height = 9

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Substituting these values into the formula, we get: 559π = π (16.8) (s) dividing both sides by π (16.8), we get: s = (559π)/ (π (16.8)) = 20.9 cm (rounded to the nearest tenth) Therefore, the slant height to the nearest tenth of a centimeter is 20.9 cm.

The Gram-Schmidt process involves taking the given vectors and constructing a new set of orthonormal vectors that span the same subspace.

The given vector is u = [-6, 1, 3] and the given subspace is spanned by the vectors v = [0, 1, 3] and w = [0, 4, 1]. The projection of vector u onto the subspace spanned by v and w is given by:

First, we will find a basis for the subspace spanned by v and w using the Gram-Schmidt process:

Normalize v to get the first basis vector

u1:u1 = v / ||v||

= [0, 1, 3] / √(0²+1²+3²)

= [0, 1/√10, 3/√10]

Next, we need to find the w projection onto the span of u1. The projection of w onto u1 is given by:

proju1w = (w⋅u1)u1

= ([0, 4, 1]⋅[0, 1/√10, 3/√10])([0, 1/√10, 3/√10])

= (4/√10)([0, 1/√10, 3/√10])

= [0, 4/10, 12/10]

= [0, 2/5, 6/5]

Normalize the projection of w to get the second basis vector

u2:u2 = proju1w / ||proju1w||

= [0, 2/5, 6/5] / √(0²+(2/5)²+(6/5)²)

= [0, 2/√65, 6/√65]

Now, we can express any vector in the subspace spanned by v and w as a linear combination of u1 and u2. To find the projection of u onto this subspace, we need to find the coefficients of this linear combination:

u = c1u1 + c2u2

=> [u1, u2][c1, c2]T

= projspan{v,w}u[u1, u2][c1, c2]T

= [-6, 1, 3]c1u1 + c2u2

= [-6, 1, 3]

Since u1 and u2 are orthonormal, we can find c1 and c2 as follows:

c1 = (u⋅u1) = [-6, 1, 3]⋅[0, 1/√10, 3/√10]

= 1/√10c2

= (u⋅u2)

= [-6, 1, 3]⋅[0, 2/√65, 6/√65]

= 18/√65

Therefore, the projection of u onto the subspace spanned by v and w is:

projspan{v,w}u = c1u1 + c2u2

= (1/√10)[0, 1, 3] + (18/√65)

= [0, 2, 6]

In finding the projection of a vector onto a subspace spanned by two or more vectors, one of the methods used is the Gram-Schmidt process, which involves constructing an orthonormal basis for the subspace. An orthonormal basis is a set of vectors that are orthogonal to each other (i.e., their dot product is zero) and have a magnitude of one.

To find the second vector, we take the projection of the second given vector onto the span of the first vector and subtract this projection from the second vector. This gives us an orthogonal vector to the first vector, but it may not be normalized.

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The orthogonal projection of vector u onto the subspace spanned by vectors v and w is approximately [0, 0.94, 0.35].

We have,

To find the orthogonal projection of vector u onto the subspace spanned by vectors v and w in the 3-dimensional vector space R³, we can use the formula:

Proj_vw(u) = ((u · v) / (v · v)) * v + ((u · w) / (w · w)) * w

Given the values for vectors u, v, and w as follows:

u = [-6, 1, 3]

v = [0, 4, 1]

w = [3, 0, 4]

We can calculate the orthogonal projection as follows:

Step 1: Calculate dot products:

u · v = (-6 * 0) + (1 * 4) + (3 * 1) = 4

u · w = (-6 * 3) + (1 * 0) + (3 * 4) = 6

v · v = (0 * 0) + (4 * 4) + (1 * 1) = 17

w · w = (3 * 3) + (0 * 0) + (4 * 4) = 25

Step 2: Calculate scalar components:

((u · v) / (v · v)) = 4 / 17

((u · w) / (w · w)) = 6 / 25

Step 3: Calculate the orthogonal projection:

Proj_vw(u) = ((4 / 17) * v) + ((6 / 25) * w)

Finally, substitute the values of v and w:

Proj_vw(u) = ((4 / 17) * [0, 4, 1]) + ((6 / 25) * [3, 0, 4])

Simplifying the expression, we find:

Proj_vw(u) ≈ [0, 0.94, 0.35]

Therefore,

The orthogonal projection of vector u onto the subspace spanned by vectors v and w is approximately [0, 0.94, 0.35].

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The probability that an arriving student will find at least one other student waiting in line is approximately 0.167.

To solve this problem, we'll use the M/M/1 queueing model with Poisson arrivals and exponential service times. Let's calculate the required values: (a) Percentage of time Judy is idle: The utilization of the system (ρ) is the ratio of the average service time to the average interarrival time. In this case, the average service time is 10 minutes, and the average interarrival time is 12 minutes. Utilization (ρ) = Average service time / Average interarrival time = 10 / 12 = 5/6 ≈ 0.8333

The percentage of time Judy is idle is given by (1 - ρ) multiplied by 100: Idle percentage = (1 - 0.8333) * 100 ≈ 16.67%. Therefore, Judy is idle approximately 16.67% of the time. (b) Average waiting time for a student:

The average waiting time in a queue (Wq) can be calculated using Little's Law: Wq = Lq / λ, where Lq is the average number of customers in the queue and λ is the arrival rate. In this case, λ (arrival rate) = 1 customer per 12 minutes, and Lq can be calculated using the queuing formula: Lq = ρ^2 / (1 - ρ). Plugging in the values: Lq = (5/6)^2 / (1 - 5/6) = 25/6 ≈ 4.17 customers Wq = Lq / λ = 4.17 / (1/12) = 50 minutes. Therefore, on average, a student spends approximately 50 minutes waiting in line.

(c) Average length of the line: The average number of customers in the system (L) can be calculated using Little's Law: L = λ * W, where W is the average time a customer spends in the system. In this case, λ (arrival rate) = 1 customer per 12 minutes, and W can be calculated as W = Wq + 1/μ, where μ is the service rate (1/10 customers per minute). Plugging in the values: W = 50 + 1/ (1/10) = 50 + 10 = 60 minutes. L = λ * W = (1/12) * 60 = 5 customers. Therefore, on average, the line consists of approximately 5 customers.

(d) Probability of finding at least one student waiting in line: The probability that an arriving student finds at least one other student waiting in line is equal to the probability that the system is not empty. The probability that the system is not empty (P0) can be calculated using the formula: P0 = 1 - ρ, where ρ is the utilization. Plugging in the values:

P0 = 1 - 0.8333 ≈ 0.1667. Therefore, the probability that an arriving student will find at least one other student waiting in line is approximately 0.167.

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The height of the ladder is 37.85 ft.

What is Trigonometric ratios?

Trigonometric ratios are ratios that relate the sides of a right triangle to its angles. There are three primary trigonometric ratios, which are commonly referred to as "trig ratios". These ratios are:

From trigonometric ratios, tan(A) = opposite / adjacent.

Here we have a ladder leans against the side of a house.

The angle of elevation of the ladder is 65° when the bottom of the ladder is 16 ft. from the side of the house.

This scenario can be converted as right angle triangle as shown in the picture

As we know from Triagonometric ratios

=> tan(A) = opposite / adjacent

=> tan (65) = L/16  

tan(A) = opposite / adjacent

tan(65) = H / 16

H = 16 tan(65)

H = 16 (2.1445)

H = 34.31 ft

Using the Pythagorean formula,

L² = H² + 16²

L² = (34.31)² + (16)²

L² = 1433.36

L ≈ 37.85 ft

Therefore, the height of the ladder is 37.85 ft.

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Answer:

C = 7.72 ~ 7.7

Step-by-step explanation:

So when you solve this equetion you must 1st find x then c

we can find x by using cos(60)

cos(60) = x/14

x = cos(60) × 14

x = 1/2 ×14

x = 7

so after we find x we are going to solve c by using cos (25)

cos (25) = X/C = 7/c

cos(25) × C = 7

C = 7/cos (25)

C = 7.72 ~ 7.7

so the solution is 7.7

Answer:

Step-by-step explanation:

is this in college?

The solution to the system of equations is:

(x, y) = (1, 2), (1, -2), (-1, 2), (-1, -2)

We have,

To solve the system of equations:

3x² + y² = 7 ---(1)

2x² - y² = -2 ---(2)

We can eliminate the variable y by adding equation (1) and equation (2):

(3x² + y²) + (2x² - y²) = 7 + (-2)

Simplifying the equation, we get:

5x² = 5

Dividing both sides of the equation by 5:

x² = 1

Taking the square root of both sides:

x = ±1

Now, we substitute the value of x back into one of the original equations (let's use equation (1)):

3x² + y² = 7

For x = 1:

3(1)² + y² = 7

3 + y² = 7

y² = 7 - 3

y² = 4

y = ±2

For x = -1:

3(-1)² + y² = 7

3 + y² = 7

y² = 7 - 3

y² = 4

y = ±2

Therefore,

The solution to the system of equations is:

(x, y) = (1, 2), (1, -2), (-1, 2), (-1, -2)

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Considering that the function spikes to infinity both to the left of x = 0 and to the right of x = 0, the function has an infinite discontinuity.

A function is said to not be continuous at a point if the function is not defined for a function. For example, in a fraction, if the denominator cannot be simplified, there is a discontinuity at the zeroes of the denominator.

In this problem, the function is defined by:

f(x) = -4/x².

The zero of the denominator is:

x² = 0 -> x = 0.

Looking for both sides of x = 0, we have that:

\(\lim_{x \rightarrow 0^-} = \lim_{x \rightarrow 0^+} = \infty\)

Hence, the function has an infinite discontinuity.

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Answer:

s = 1/2

Step-by-step explanation:

Solve for s:

4 s^2 - 4 s + 1 = 0

Divide both sides by 4:

s^2 - s + 1/4 = 0

Write the left hand side as a square:

(s - 1/2)^2 = 0

Take the square root of both sides:

s - 1/2 = 0

Add 1/2 to both sides:

Answer: s = 1/2

Answer:

s = \(\frac{1}{2}\)

Step-by-step explanation:

To find the zero set the quadratic equal to zero, that is

4s² - 4s + 1 = 0 ← in standard form

(2s - 1)(2s- 1) = 0 ← in factored form

(2s - 1)² = 0 ← perfect square, thus

2s - 1 = 0 ⇒ 2s = 1 ⇒ s = \(\frac{1}{2}\) ← is the zero

To find the dimensions that yield the maximum area for a rectangular room with a fixed perimeter of 280ft, we can write a quadratic function for the area and determine the vertex.

Let's assume the length of the room is x and the width is y. The perimeter of a rectangle is given by the formula P = 2(x + y), so in this case, we have 2(x + y) = 280ft, which simplifies to x + y = 140ft.

The area of a rectangle is given by the formula A = xy. We can express one variable in terms of the other by substituting y = 140 - x into the area formula, giving A = x(140 - x) = 140x - x^2.

This equation represents a quadratic function, where the coefficient of the quadratic term is negative. The vertex of a quadratic function occurs at the axis of symmetry, which can be found using the formula x = -b/(2a). In this case, a = -1 and b = 140, so the axis of symmetry is x = -140/(2*(-1)) = 70ft.

To find the corresponding width, we substitute this value back into x + y = 140ft: 70 + y = 140, which gives y = 70ft.

Therefore, the dimensions that yield the maximum area for the rectangular room are 70ft for the length and 70ft for the width.

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The area of the shaded region is 34%.

Given graph depicts the IQ scores of adults, with a mean of 100 and a standard deviation of 15.

The probability density function of the normal distribution is given

byf(x) = (1/σ√(2π)) * e^[-(x-μ)²/(2σ²)]

Here, x = IQ scores of adults,

μ = Mean = 100σ = Standard deviation = 15

The area of the shaded region is the area between the Z-score values of -1 and 1. Since, we know that the mean of the normal distribution is 100, we can use the formula for the Z-score,

Z = (X - μ) / σ

⇒ Z = (100 - 100) / 15

= 0

Therefore, the Z-score of X = 100 is 0.

Also, we can use the empirical rule to find the percentage of data that falls within 1 standard deviation of the mean.

The empirical rule states that, For the normal distribution,68% of data falls within 1 standard deviation of the mean.

Using this, we can find the area of the shaded region.Area of the shaded region = [68/2]% = 34%

Therefore, the area of the shaded region is 34%.

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Answer:

No,the number isn't a perfect square

Step-by-step explanation:

Taking the prime factor of the number, each factor can be raised to the power of 2 except 3

The angle of elevation from the stake up to the tent pole is, 54.55°.

We know a right-angled triangle has three sides they are -: Hypotenuse,

Opposite and Adjacent.

We can remember SOH CAH TOA which is,

sin = opposite/hypotenuse, cos = adjecen/hypotenuse and

tan = opposite/adjacent.

Given, A 19 ft rope is tied from the top of a tent pole, It is the hypotenuse length.

Also, The state is 11 feet away and it is the adjacent length.

We know, cos = adjacent/hypotenuse.

cos = 11/19.

Ф = cos⁻¹(0.58).

Ф = 54.55°.

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The explicit formula for the sequence an = 1-en-1 nten is an = 1 - e^(n-1) * (n-1) * e.

Alternatively, if we consider the sequence an = 1-en-1 2+en, the explicit formula would be an = 1 - e^(n-1) * (n-1) * e + e^(n-1) * (n+1) * e. Lastly, if we consider the sequence an = 1-en 2+en, the explicit formula would be an = 1 - e^n * n * e + e^(n-1) * (n+2) * e.

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Answer:

um.... 17??????????????????

Answer:

HDHEHFGEWGDYDGYDHWQFSWDWSQWYUWEHWW8WDWD

Step-by-step explanation:BHDHEJKNHRGHRFYRHDHNHNEBFDYHFBVUUUFHECHFHVYGDFRFHRHFFRUFYYRFFRYFYFYRFREYRYFGFWYBGT

Answer:

What is 5% of 45.22?

That would be about 2.3

Now times that by 4 then there is your answer :)

The required product are as follows,
A: -4p²q²-3pq = 4c²[1 - 3c²]     B: 4c²-12c⁴ = 4c²[1 - 3c²]    
C: 3m²+9m³ = 3m²{1 + 3m}       D: c³+c⁷ = c³[1 + c⁴]
E: 24x⁴ - 16x⁸ = 8x⁴[3 - 2x²]

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,
(a)
Given expression,
-4p²q²-3pq = -pq (4pq + 3)

Similarly,
B: 4c²-12c⁴ = 4c²[1 - 3c²]

C: 3m²+9m³ = 3m²{1 + 3m}

D: c³+c⁷ = c³[1 + c⁴]

E: 24x⁴ - 16x⁸ = 8x⁴[3 - 2x²]

Thus, the required products has be illustrated above.

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Answer:

D.$9.78

Step-by-step explanation:

20% of $8.15 $8.15 + $1.63

=8.15 x 20/100 =$9.78

=$1.63

Answer:

9x-5

Step-by-step explanation:

(4x - 6) + (5x + 1)

Add like terms

4x+5x=9x

-6+1=-5

Combine

9x-5

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If g(x) is defined for all real numbers, then the domain of (f+g)(x) is also the set of all real numbers.

The domain of (f+g)(x) can be determined by finding the common domain of f(x) and g(x).

If f(x) and g(x) have different domains, then the domain of their sum (f+g)(x) is restricted to the intersection of their domains.

In this case, the domain of g(x) is not given. Therefore, we cannot determine the domain of (f+g)(x).However, if we assume that the domain of g(x) is the set of all real numbers, then the domain of (f+g)(x) would be the set of all real numbers, since the domain of f(x) is also the set of all real numbers.

Here is how we can write the answer using HTML format: To find the domain of (f+g)(x), we need to determine the common domain of f(x) and g(x).

Otherwise, we cannot determine the domain of (f+g)(x) without more information about g(x).

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Answer:

1 is the answer

Step-by-step explanation:

When coordinates are reflected over the x axis, the x coordinates stay the same but the y coordinates becomes either negative, if they were positive, or positive, if they had orginally been negative.

If this doesn't make sense i'l try to rephrase it, hope it helps though :)

Answer:

x =0 ,x =7

Step-by-step explanation:

Method 1 ; Factoring

\(9x^2 -63x =0\\\\\mathrm{Solve\:by\:factoring}\\\mathrm{Factor\:}9x^2-63x:\quad 9x\left(x-7\right)\\\\9x\left(x-7\right)=0\\\\\mathrm{Using\:the\:Zero\:Factor\:Principle:}\\if\:ab=0\:\mathrm{then}\:a=0\:\mathrm{or}\:b=0\:\left(\mathrm{or\:both}\:a=0\:\mathrm{and}\:b=0\right)\\\left(\mathrm{or\:both}\:a=0\:\mathrm{and}\:b=0\right)\\\\x =0\\\mathrm{Solve\:}\:x-7=0:\quad x=7\\x=0,\:x=7\)

Method 2 ; Completing the square

\(9x^2-63x=0\\\\\mathrm{Divide\:both\:sides\:by\:}9\\\frac{9x^2-63x}{9}=\frac{0}{9}\\\\x^2-7x=0\\\mathrm{Solve\:by\:completing\:the\:square}\\\mathrm{Write\:equation\:in\:the\:form:\:\:}\\x^2+2ax+a^2=\left(x+a\right)^2\\\\\mathrm{Solve\:for\:}a,\:2ax=-7x:\quad a=-\frac{7}{2}\\\\\mathrm{Add\:}a^2=\left(-\frac{7}{2}\right)^2\mathrm{\:to\:both\:sides}\\x^2-7x+\left(-\frac{7}{2}\right)^2=0+\left(-\frac{7}{2}\right)^2\\\\\mathrm{Simplify\:}\\0+\left(-\frac{7}{2}\right)^2:\quad \frac{49}{4}\)

\(x^2-7x+\left(-\frac{7}{2}\right)^2=\frac{49}{4}\\\\\mathrm{Complete\:the\:square}\\\left(x-\frac{7}{2}\right)^2=\frac{49}{4}\\\\\mathrm{For\:}f^2\left(x\right)=a\mathrm{\:the\:solutions\:are\:}f\left(x\right)=\sqrt{a},\:-\sqrt{a}\\\\\mathrm{Solve\:}\:x-\frac{7}{2}=\sqrt{\frac{49}{4}}:\quad x=7\\\\\mathrm{Solve\:}\:x-\frac{7}{2}=-\sqrt{\frac{49}{4}}:\quad x=0\\\\x=7,\:x=0\)

Answer:

y=2x+1

Step-by-step explanation:

y= mx + c

c= 1

m= slope which is -2

y=2x+1

Answer:

True

Step-by-step explanation:

The wording of the question is a little tricky, but here's what I think.

If a function f(x) is not defined at a point c, then the function has a discontinuity at that point. In order for a function to be continuous on an interval, it must be defined and have no abrupt changes or jumps within that interval. Since f(x) is not defined at c, it violates the condition of continuity, and therefore f(x) cannot be continuous on any interval that includes c.

The given statement "If f(x) is not defined at c, then f(x) cannot be continuous on any interval." is false because it does not automatically mean that f(x) cannot be continuous on any interval.

Continuity of a function depends on the behavior of the function around the point of interest, rather than just the absence of a definition at a single point. A function can still be continuous on an interval except at the specific point where it is not defined.

For example, consider the function f(x) = 1/x. This function is not defined at x = 0, but it is continuous on any interval that does not include x = 0. This is because f(x) approaches positive or negative infinity as x approaches 0 from the left or right side, respectively, indicating that there is no abrupt jump or discontinuity.

In general, the continuity of a function is determined by its behavior around a point, including its limit as x approaches that point. The absence of a definition at a single point does not automatically imply that the function cannot be continuous on any interval.

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Answer:

The cookies cost $6 each, and $12 altogether.

Step-by-step explanation:

50-38=12, or x

Solution:

Given:

A word problem.

To solve the question, we develop the statements into mathematical equations.

Hence,

Let the two numbers be represented by x and y

where;

x is one number

y is another number

Solving both equations generated simultaneously, using the elimination method, we subtract equation (1) from equation (2).

That is equation (2) - equation (1);

Substituting the value of y gotten into equation (2) to solve for x,

Hence, the solution to the equations is;

Therefore, the numbers are 10 and 4.

There is a 4/9 probability that Ava and David will be on the same team.

There are a total of 10 players, and we need to randomly select two teams of five players each. The total number of ways to select the two teams is given by the combination formula:

C(10,5) * C(5,5) = 252 * 1 = 252

This is because we first choose 5 players for the green team from 10 players, and then choose the remaining 5 players for the purple team from the remaining 5 players.

Now, we need to find the number of ways in which Ava and David can be on the same team. There are two cases to consider: either they are both on the green team, or they are both on the purple team.

Case 1: Ava and David on green team

To calculate the number of ways in which Ava and David can be on the green team, we first choose 3 players from the remaining 8 players to join them. This can be done in C(8,3) ways. So the total number of ways in which Ava and David can be on the green team is:

C(8,3) = 56

Case 2: Ava and David on purple team

The number of ways in which Ava and David can be on the purple team is the same as the number of ways in which they can be on the green team, which is 56.

So the total number of ways in which Ava and David can be on the same team is 56 + 56 = 112. Therefore, the probability that Ava and David will be on the same team is:

P(Ava and David on same team) = 112/252 = 4/9

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average rate of change = -1

Average rate of change formula:

The range is from 1 to 3.

b= 3, a = 1

From the given function, x is greater than 1 when f(x) = -x + 2

when x = 3

f(3) = -3 + 2 = -1

when x = 1

f(1) = -1 + 2 = 1