What are the length and width of a rectangle if the length is
3 inches longer than twice the width and the area of the
rectangle is 5 in2?

Answer: Scott - 18, Kaitlin - 9, Bill - 54

Step-by-step explanation:

Let's view this as units. There are a total of 81 units. If we solve for Scott's orders:

Kaitlin served 9 fewer orders than Scott. We add 9 to 81 --> 9 + 81 = 90.

Now if we view Scott's orders as one unit:

Scott - 1 unit

Kaitlin (after adding 9) - 1 unit

Bill - 3 units (he served 3 times as many orders as Scott)

In total there are 3 + 1 + 1 = 5 units.

90 / 5 = 18.

Therefore, Scott served 18 orders.

18 - 9 = 9.

Kaitlin served 9 orders.

18 x 3 = 54.

Bill served 54 orders.

And to check our answer, 54 + 18 + 9 is 81.

The equation for the plane that contains the line and is perpendicular to the plane 2 · x + y - 3 · z = - 4 is - 10 · x + 17 · y - z = 76.

From the equation of the plane 2 · x + y - 3 · z = - 4 we know that its normal vector is (2, 1, - 3). Hence, we have found two direction vectors and already know a point, which are part of this parametric equation:

(x, y, z) = (- 3, 4, 5) + t · (3, 2, 4) + u · (2, 1, - 3)     (1)

The normal vector of the plane is equal to the cross product of the direction vectors of (1):

[tex]\vec n = (3, 2, 4) \,\times\,(2, 1, -3)[/tex]

[tex]\vec n = \left|\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\3&2&4\\2&1&-3\end{array}\right|[/tex]

[tex]\vec n = (-6-4,8+9, 3-4)[/tex]

[tex]\vec n = (-10, 17, -1)[/tex]

Since the plane contains the point (x, y, z) = (- 3, 4, 5), then we find that the independent constant of the equation of the plane is:

- 10 · x + 17 · y - z = k

- 10 · (- 3) + 17 · 4 - 5 = k

30 + 51 - 5 = k

k = 76

The equation for the plane that contains the line and is perpendicular to the plane 2 · x + y - 3 · z = - 4 is - 10 · x + 17 · y - z = 76.

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The length if the upper portion of the stand is 33. 56 inches

The formula for the finding the length is given thus;

Total height = length of upper portion + length of base

Where

Length of base = 257/16

Total height = 591/2

Length of upper portion = total height - length of base

= 118/2 - 407/16

= 59 - 25. 44

= 33. 56 inches

Thus, the length if the upper portion of the stand is 33. 56 inches

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The given point on the midline of (π, -4), and the minimum point of (π/4, -5.5), give the exact expression for f(x) as; f(x) = 1.5•sin((2/3)•(x - π)) - 4.

The amplitude, a = The difference between the y-values at the minimum point and the midline

Therefore;

a = -4 - (-5.5) = 1.5

d = The difference between the y-values of the midline and the x-axis

Therefore;

d = -4 - 0 = -4

The given function is presented as follows;

f(x) = a•sin(b•x + c) + d

Which gives;

-4 = 1.5•sin(b•π + c) - 4

sin(b•π + c) = 0

(b•π + c) = 0

-c/b = π

c = -b•π

Therefore;

-5.5 = 1.5•sin(b•π/4 + c) - 4

-1.5 = 1.5•sin(b•π/4 + c)

-1 = sin(b•π/4 - b•π) = sin(-3•b•π/4)

-3•b•π/4 = arcsine (-1) = -π/2

-3•b•π/4 = -π/2

3•b/4 = 1/2

b = 2/3

c = -(2/3)•π

Therefore, f(x) = a•sin(b•x + c) + d, gives;

f(x) = 1.5•sin((2/3)•x - (2/3)•π) - 4

By simplification, the exact expression for f(x) is therefore;

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Using the normal distribution, the areas to the left are given as follows:

a) 0.7910.

b) 0.6664.

c) 0.3707.

d) 0.8508.

The z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

Hence:

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The test statistic, rounded to 2 decimal places is equal to

For this sample, the hypothesis is given by:

H₀: μ₁ ≤ μ₂

H₁: μ₁ > μ₂

Assuming this sample has a normal distribution, we would use a pooled z-test to determine the value of the test statistic:

Substituting the given parameters into the formula, we have;

[tex]z = \frac{\frac{96}{150} \;-\;0.63}{\sqrt{0.63\; +\; \frac{1\;-\;0.63}{150}} }\\\\z = \frac{0.64 \;-\;0.63}{\sqrt{0.63\; +\; \frac{0.37}{150}}}\\\\z = \frac{0.01}{\sqrt{0.63\; +\; 0.0025}}\\\\z = \frac{0.01}{\sqrt{0.6325}}\\\\[/tex]

z = 0.01/0.7953

z = 0.013.

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The value of r If r varies directly with the square of m and inversely with s is 240

Suppose r varies directly with the square of m and inversely with s.

If r varies directly with the square of m and inversely with s, then;

r = km²/s

Given that R = 15 when m = 6 s= 4

15 = k(6)²/4

60 = 36k

k = 60/36

k = 5/3

If the value of m = 24 and s = 4, hence;

r = 5/3*24²/4

r = 240

Hence the value of r If r varies directly with the square of m and inversely with s is 240

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The number of ways to forma a square by connecting the ends of some of the line segments is 3, 024 ways

From the given information, we should use the formula for permutation without repetition.

The formula is given as;

Permutation = [tex]\frac{n!}{n -r !}[/tex]

Where n = number of set = 9

r = 4, this is so because, the sides of a square a four

Permutation = [tex]\frac{9!}{9 - 4!}[/tex]

Permutation = [tex]\frac{9!}{5!}[/tex]

Permutation = [tex]\frac{362, 880}{120}[/tex]

Permutation = 3, 024 ways

Thus, the number of ways to forma a square by connecting the ends of some of the line segments is 3, 024 ways

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Answer: The largest he can possibly build is x = 14

Step-by-step explanation:

The line perpendicular to the equation and has the same y-intercept with another is y = - 4 / 3 x + 4.

The equation of a line can be describe as follows:

y = mx + b

where

Therefore, lines that a perpendicular follows the rule below:

m₁m₂ = -1

Hence,

y + 6 = 3 / 4 (x - 2)

y + 6 = 3 / 4 x - 6 / 4

y =  3 / 4 x - 6 / 4 - 6

y = 3 / 4 x - 30 / 4

y = 3 / 4 x - 15 / 2

Hence,

3 / 4 m₂ = -1

m₂ = - 4 / 3

Therefore,

slope of the line is - 4 / 3

Let's find the y-intercept of the second line

4x + 5y -20 = 0

5y = -4x + 20

y = -4 / 5 x + 4

y-intercept  = 4

Therefore, the line perpendicular to the equation and has the same y-intercept with another is y = - 4 / 3 x + 4.

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Answer:

Step-by-step explanation:

Shift right

f(x)=√x-4.7

f(x)=√x-25

f(x)=√x-11

Shift left

f(x)=√x+0.8

f(x)=√x+6

The shortest distance from A to B in the diagram is 17 m.

The shortest distance from D to B can be found as follows:

using Pythagoras theorem,

c² = a² + b²

where

Therefore,

DB² = 15² + 8²

DB² = 225 + 64

DB² = 289

DB = √289

DB = 17 m

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Answer:

B, 17 m

Step-by-step explanation:

Founders 23

The length of AC is 12 units . Option D

The formula for area of a kite is given thus;

Area = [tex]\frac{p * q}{2}[/tex]

Where p and q are the lengths

Area of kite = 48 square units

BD, q = 8 units

AC , p = unknown

Substitute the values

[tex]48 = \frac{AC * 8}{2}[/tex]

Cross multiply

[tex]AC * 8 = 48 * 2[/tex]

[tex]8AC = 96[/tex]

[tex]AC = \frac{96}{8}[/tex]

[tex]AC = 12[/tex] units

Thus, the length of AC is 12 units . Option D

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Answer:option 4

Step-by-step explanation:

The dairy farmer would mix 1120 pounds of 75% protein supplement and 280 pounds of 50% protein ration to make 1400 pounds of a high grade 70% protein ration.

An equation is an expression that shows the relationship between two or more variables and numbers.

Let x represent the amount of 75% protein supplement and y represent the amount of 50% protein, hence:

x + y = 1400     (1)

Also:

0.75x + 0.5y = 0.7(1400)     (2)

Hence:

x = 1120, y = 280

The dairy farmer would mix 1120 pounds of 75% protein supplement and 280 pounds of 50% protein ration to make 1400 pounds of a high grade 70% protein ration.

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The sum of the arithmetic series is 325

The given parameters are

a1 = 45

an = 85

n = 5

The formula for the sum of arithmetic series is:

[tex]S_n = \frac n2 * (a_1 + a_n)[/tex]

Substitute the values of a1, an and n in the above equation

[tex]S_5 = \frac 52 * (45 + 85)[/tex]

Add 45 and 85

[tex]S_5 = \frac 52 * 130[/tex]

Divide 130 by 2

[tex]S_5 = 5 * 65[/tex]

Multiply

[tex]S_5 = 325[/tex]

Hence, the sum of the arithmetic series is 325

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350 grams of magic steel and 350 grams of dragon alloy #33 is needed to make 700 grams of Dragon alloy #18.

An equation is an expression that shows the relationship between two or more numbers and variables.

Let x represent the amount of magic steel and y represent the amount of alloy #33, hence:

(0 * x) + (y * 0.28) = 700(1 - 0.86)   (1)

Also:

x + (1 - 0.28)y = 0.86(700)    (2)

From both equations:

x = 350, y = 350

350 grams of magic steel and 350 grams of dragon alloy #33 is needed to make 700 grams of Dragon alloy #18.

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The maximum value of the function 7 - 2x - 3x² exists 22/3.

Given: 7 - 2x - 3x²

We must begin by estimating the first derivative:

Differentiating the equation above, we have:

dy/dx = 0 - 2 - 6x = - 2 - 6x

We know that dy/dx = 0 at maxima and minima

Therefore, we contain, - 2 - 6x = 0

6x = - 2

x = - 1/3

Substituting x = - 3 back into the equation of the curve yields the following result:

y = 7 - 2(-1/3) - 3(-1/3)² = 22/3

y = 22/3

Therefore, 22/3 exists the maximum value of y.

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The equivalent expressions are 6y + 4 and 11y + 16

Expression (A)

We have:

2y + 4(y + 1)

Open the bracket

2y + 4y + 4

Evaluate the like terms

6y + 4

Expression (B)

We have:

9y + 6 + 2(5 + y)

Open the bracket

9y + 6 + 10 + 2y

Evaluate the like terms

11y + 16

Hence, the equivalent expressions are 6y + 4 and 11y + 16

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The probability that the ball Deborah picks is green is; 3/5.

Since the bag initially contains; 3 red balls, 4 green balls, and 2 yellow balls, after Deborah pulls out the 4 balls: red, red, green, yellow.

The balls remaining are; 1 red ball, 3 green and 1 yellow.

Hence, the probability that the final ball she picks is green is; 3/(3+1+1) = 3/5.

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Answer:

B, -1.7

Step-by-step explanation:

-10.2 divided by 6 is -1.7.

I hope this helps!

Answer:

[tex]\dfrac{3^{13}}{2}[/tex]

Step-by-step explanation:

Given expression:

[tex]\dfrac{3^5 \cdot 2^4 \cdot 2^1 \cdot 3^6 \cdot (3^2)^4}{(2^3)^2 \cdot 3^6}[/tex]

[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]

[tex]\implies \dfrac{3^5 \cdot 2^4 \cdot 2^1 \cdot 3^6 \cdot 3^8}{2^6 \cdot 3^6}[/tex]

[tex]\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^{b+c}:[/tex]

[tex]\implies \dfrac{3^{(5+6+8)} \cdot 2^{(4+1)}}{2^6 \cdot 3^6}[/tex]

[tex]\implies \dfrac{3^{19} \cdot 2^{5}}{2^6 \cdot 3^6}[/tex]

[tex]\implies \dfrac{3^{19} \cdot 2^{5}}{3^6 \cdot 2^6}[/tex]

[tex]\textsf{Apply exponent rule} \quad \dfrac{a^b}{a^c}=a^{b-c}:[/tex]

[tex]\implies 3^{(19-6)} \cdot 2^{(5-6)}[/tex]

[tex]\implies 3^{13} \cdot 2^{-1}[/tex]

[tex]\textsf{Apply exponent rule} \quad a^{-n}=\dfrac{1}{a^n}:[/tex]

[tex]\implies \dfrac{3^{13}}{2^1}[/tex]

[tex]\textsf{Apply exponent rule} \quad a^1=a:[/tex]

[tex]\implies \dfrac{3^{13}}{2}[/tex]

Answer:

The result is 3¹³ /2

Step-by-step explanation:

Greetings

Answer:

  h > 5

  h < 0

Step-by-step explanation:

We suppose that a "Special Solution" inequality is one that is technically incorrect, but is written using a compact "shorthand" form.

As written, the inequality claims that 17 < 2, which is false. There can be no value of the variable that will make this true. We presume the intended meaning is ...

  17 < 3h +2   or   3h +2 < 2

Subtracting 2 from the inequality gives ...

  15 < 3h < 0

Dividing by 3 gives ...

  5 < h < 0

There is no value of h that is both greater than 5 and less than 0, so we presume this means the OR (union) of the solution sets ...

  h > 5

  h < 0

28 miles upstream requires 28/7 = 4 hours.

28 miles downstream requires 28/21 = 4/3 hours, or 1 hour and 20 minutes.

Apply the distance formula:

d = rt

where

d is distance

r is rate or speed

t is time

21 = t*(r+7)  = rt+ 7t

7 = t *(r-7)   = rt - 7t

            28 = 2rt

             14 = rt

             21 = rt + 7t

             21 = 14 + 7t

               7 = 7t

                 t=1

              21 = (r+7)*1 = r+7

                r = 14

The speed of the boat is 14 mph. With the current (downstream), the speed is 14+7 = 21 mph, so 21 miles are traveled in 1 hour. Against the current (upstream), the speed is 14-7 = 7 mph, which means 7 miles are traveled in the same 1-hour time frame.

28 miles downstream requires 28/21 = 4/3 hours, or 1 hour and 20 minutes.

28 miles upstream requires 28/7 = 4 hours.

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Answer:

B

Step-by-step explanation:

because at origin they are both positive and when it rotates about 270 both will change to negative

Answer:

The transformation being described is from g(x)=x2 g ( x ) = x 2 to f(x)=x2 f ( x ) = x 2 . The horizontal shift depends on the value of h . The horizontal shift is described as: f(x)=f(x+h) f ( x ) = f ( x + h ) - The graph is shifted to the left h units

Answer:

Greetings!

The answer for the first figure is attached but the second diagram is not clear.

Also use the image i attached horizontally.

and also it TD subject not Mathematics.

Using the t-distribution, the 96% confidence interval is given as follows:

(18.96, 21.04).

The confidence interval is:

[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]

In which:

The critical value, using a t-distribution calculator, for a two-tailed 96% confidence interval, with 100 - 1 = 99 df, is t = 2.0812.

The parameters are given as follows:

[tex]\overline{x} = 20, s = 5, n = 100[/tex]

Hence the bounds of the interval are:

[tex]\overline{x} - t\frac{s}{\sqrt{n}} = 20 - 2.0812\frac{5}{\sqrt{100}} = 18.96[/tex]

[tex]\overline{x} + t\frac{s}{\sqrt{n}} = 20 + 2.0812\frac{5}{\sqrt{100}} = 21.04[/tex]

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Answer:

15/16

Step-by-step explanation:

because 2/3have Becky