The hypotenuse of a certain right triangle is twice the length of the
shorter leg. The shorter leg is 4 cm.

Answer:

14 mins

Step-by-step explanation:

2/6 = x/42

6x=84

x=14

Suppose you are a small business owner who sells handmade crafts. You have a budget of $100 to purchase materials for your next batch of products. You know that each craft requires some amount of materials, which costs $8 per unit. Additionally, you will need to pay a fixed cost of $14 for other expenses related to production and shipping.

To make a profit, you must ensure that the cost of materials and fixed expenses does not exceed the $100 budget. Therefore, you can write an inequality to represent this situation:

8x + 14 < 100

Here, x represents the number of units of materials needed for each craft. The inequality states that the total cost of materials (8x) plus fixed expenses ($14) must be less than $100.

To solve this inequality, you can subtract 14 from both sides:

8x < 86

Finally, you can divide both sides by 8:

x < 10.75

This means that for each craft, you can use no more than 10.75 units of materials in order to stay within budget and make a profit.

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The volume of the clay used to make the vase is 36 in³

To determine the volume of clay that was used for the vase, we need to consider that;

The formula for the volume of a rectangular prism is expressed as;

Volume = lwh

Such that;

For the bigger prism, we have;

Volume = 3 × 3 × 5

Multiply the values

Volume = 45 in³

For the hole, we have;

Volume = 4 × 3/2 × 3/2

Multiply the values

Volume = 9 in³

Then, the volume of clay used for the vase = 45 - 9 = 36 in³

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Ina rows at a rate of 6 mph in still water, and the current flows at a rate of 2 mph.

Distance is the overall length of a journey that a person, item, or vehicle takes to get from one place to another. Due to its scalar nature, it has simply magnitude and no direction. Typically, distance is expressed in terms of metres, kilometres, miles, or feet. It is distinct from displacement, which describes the shift in an object's location from its starting point to its ending point while also taking the movement's direction into consideration.

We can start by using the formula:

distance = rate × time

Let's first consider the trip downstream. In this case, Ina's speed relative to the river current adds up to the distance traveled per unit of time. So we have:

16 = (x + y) × 2

Simplifying and solving for x + y, we get:

x + y = 8

Now let's consider the trip upstream. In this case, Ina's speed relative to the river current subtracts from the distance traveled per unit of time. So we have:

16 = (x - y) × 4

Simplifying and solving for x - y, we get:

x - y = 4

We now have two equations with two variables, so we can solve for x and y using either substitution or elimination. Let's use elimination:

(x + y) + (x - y) = 8 + 4

2x = 12

x = 6

Now we can substitute x = 6 into either of the original equations to solve for y. Let's use the first equation:

x + y = 8

6 + y = 8

y = 2

Therefore, Ina rows at a rate of 6 mph in still water, and the current flows at a rate of 2 mph.

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The interest rate of the savings account would be = 33.3%

To calculate the interest rate of the savings account, the formula for simple interest should be used. That is;

Simple interest = Principal × time × Rate/100

simple interest = 1250-312.50 = 937.5

Principal amount = $312.50

Time = 9 years

rate = ?

That is;

937.5 = 312.50 × 9 × R/100

make R the subject of formula:

R = 937.5× 100/312.50 × 9

= 93750/2812.5

= 33.3%

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To solve the problem, we need to find Julian's average speed, given that he started biking from a position behind the simulated biker at a speed of 20 km/h, and after 15 minutes, his position was reported as -214 km.

We can use the formula for average speed:

Average speed = total distance / total time

To find the total distance, we need to calculate the displacement of Julian from the initial position of -d (where d is the distance between Julian and the simulated biker when he started biking) to the position of -214 km after 15 minutes.

Displacement = final position - initial position

Displacement = (-214 km) - (-d) = d - 214 km

The total distance covered by Julian is equal to the absolute value of the displacement, since the direction of the motion does not matter when computing distance.

Total distance = |d - 214 km|

To find the total time, we need to convert 15 minutes to hours:

Total time = 15 minutes / 60 minutes/hour = 0.25 hours

Now we can substitute the values into the formula for average speed:

Average speed = total distance / total time

Average speed = |d - 214 km| / 0.25 hours

Since Julian was traveling at a constant speed of 20 km/h, we can also express the distance in terms of time:

Average speed = (20 km/h) x t / 0.25 hours

where t is the time Julian biked in hours.

Setting the two expressions for average speed equal to each other, we can solve for t:

|d - 214 km| / 0.25 hours = (20 km/h) x t / 0.25 hours

|d - 214 km| = 20 km/h x t

Solving for t:

t = |d - 214 km| / 20 km/h

Now we can substitute this expression for t into either expression for average speed:

Average speed = (20 km/h) x t / 0.25 hours

Average speed = |d - 214 km| / 0.25 hours

Substituting the expression for t:

Average speed = |d - 214 km| x 4 / |d - 214 km|

Simplifying:

Average speed = 80 km/h

Therefore, Julian's average speed so far has been 80 km/h.

The total cost for the first 25 units is given as follows:

$140.

The marginal cost function is defined as follows:

[tex]c(x) = \frac{14}{\sqrt{x}} = 14x^{-\frac{1}{2}}[/tex]

The total cost function is the integral of the marginal cost function, hence it is given as follows:

[tex]t(x) = \int c(x) dx[/tex]

[tex]t(x) = 28\sqrt{x}[/tex]

Applying the Fundamental Theorem of Calculus, the total cost from x = 0 to x = 25 is given as follows:

t(25) - t(0).

Thus:

Hence:

140 - 0 = $140.

(Applying the Fundamental Theorem of Calculus, we subtract the numeric values at each endpoint of the integral).

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There were no more books checked out in March than in April.

The pictograph provided shows the number of books checked out from the library during the months of December, January, February, March, and April. Each picture represents 5 books.

The pictograph tells us that for each of these months, 5 books were checked out. So, in March, 5 books were checked out, and in April, 5 books were also checked out.

To find the difference between the number of books checked out in March and April, we subtract the number of books checked out in April from the number of books checked out in March:

5 books (March) - 5 books (April) = 0 books

Therefore, The result shows that there were no more books checked out in March than in April. Both months had the same number of books checked out, which is 5.

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The solutions to f(x) = 0 on the interval [0, 2pi) are x = 0 and x = pi.

(a) For f(x) = 0

We set the function equal to zero and solve for x:

f(0) = sec²x - 1 = 0

sec² x = 1

Taking the square root of both sides, we get:

sec x = ±1

Recall that sec x = 1/cos x = cos⁻¹ x

Therefore

x = cos⁻¹ (1) = 0 (using calculator or table)

x = cos⁻¹(-1) = pi

Therefore, the solutions to f(x) = 0 on the interval [0, 2pi) are x = 0 and x = pi.

(b) For f(x) > 0

To get the positive values of x, we can start by factoring f(x):

f(x) = sec²x - 1 = (sec x + 1)(sec x - 1)

Since the square of the secant function is always positive, we have:

f(x) > 0 if and only if (sec x + 1)(sec x - 1) > 0

There are two cases to consider:

sec x + 1 > 0 and sec x - 1 > 0

sec x + 1 < 0 and sec x - 1 < 0

For case 1, we have:

sec x > -1 and sec x > 1

Since secant is always positive, we have sec x > 1.

For case 2, we have:

sec x < -1 and sec x < 1 (This is not possible)

(c) For f(x) < 0

By using the factored form of f(x) from part (b), we can find the values of x where f(x) is negative.

f(x) < 0 if and only if (sec x + 1)(sec x - 1) < 0

There are two cases to consider:

sec x + 1 > 0 and sec x - 1 < 0

sec x + 1 < 0 and sec x - 1 > 0

For case 1, we have:

sec x > 1 and sec x < -1 (not possible)

For case 2, we have:

sec x < -1 and sec x > 1

Since secant is always positive, this case is not possible either.

Therefore, there are no solutions to f(x) < 0 over the interval [0, 2pi).

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Answer:

-9

Step-by-step explanation: Definitely correct

A function whose graph is the graph of y = √x, but is shifted to the left 9 units is [tex]y=\sqrt{x+9}[/tex].

In Mathematics and Geometry, the translation of a geometric figure or graph to the left means subtracting a digit from the value on the x-coordinate of the pre-image;

g(x) = f(x + N)

Conversely, the translation a geometric figure upward means adding a digit to the value on the positive y-coordinate (y-axis) of the pre-image; g(x) = f(x) + N.

Based on the information provided, the transformed function should be written as follows:

y = √x

g(x) = (√x + 9)

[tex]y=\sqrt{x+9}[/tex]

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Answer:

x2+10x+25

Step-by-step explanation:

It goes like this:

X2+2*1x*5+5*5

The hexagon with sides of length 5 inches, 4 inches, 4 inches, 5 inches, 3 inches, and 3 inches has a perimeter of 24 inches.

A hexagon is a regular polygon, meaning all sides and angles are equal in size. It is a symmetrical shape, which means it can be divided into two equal halves.

To calculate the perimeter of this hexagon, we must first identify the length of each side.

All sides of the hexagon have a length of either 5 inches, 4 inches, or 3 inches, as there are two sides of each length.

The perimeter of the hexagon is the sum of the length of all its sides. Adding the length of all the sides, we get

5 + 4 + 4 + 5 + 3 + 3 = 24.

Thus, the perimeter of the hexagon is 24 inches.

Hence, the sum of the length of the sides of a hexagon is always greater than the length of any one of its sides.

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Note that 7th grade mean = 277.86

the 5th grade mean = 254.77

So th relationship between the mean is that the 7 th grade mean is higher than the 5 th grade mean .

To compute the means here   is what we did

7 th  grade mean = (1 × 10) + (1 × 11) +   (2 × 12) + (1 × 13) +   (1 × 14) + (2 × 15) + (3 × 16) + (3 × 17) + (2 × 18) + (2 × 19) +   (3 × 20) / 21

= 277.857142857

≈ 277.86

For the 5th grade mean

5th  grade mean = (1 × 8)   + (2 × 9) + (2 × 10) +   (2 × 11) + (2 × 12) + (1 × 13) + (3 × 15) + (2 ×16) + (1 × 17)  +  (2 × 18) + (1 × 19) + (1   × 20) / 26 = 12.5

= 254.769230769

≈ 254.77

This means that  trully,  the relationship between the mean is that the 7  th grade mean is higher than the 5 th grade mean.

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The number of packing materials the Amelia will need = 2.4

From the diagram given above, the formula for the volume of a right rectangular prism = Length×width ×height.

Where;

Length = 10in

Width = 6 in

Height = 8 in

The volume = 10×6×8 = 480 in³

But one bag = 200in³

The amount of bag that the music box can contain = 480/200 = 2.4

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Answer:

Step-by-step explanation:

2

2-9=-7

In negative numbers, the greater number that is negative is less than the smaller negative number. For example, -9<-8. Even though 9 is a bigger number when positive, it's the opposite for negative numbers. While there can be many numbers for k like 0,1, or 2, 1 will probably be the best answer.

Answer:

Step-by-step explanation:

[tex]\mathrm{ k - 9 < - 6}[/tex]

Add both sides by 9 :-

Simplify :-

___________________

Hope this helps!

A graph that represent the quadratic equation y = -x² + 8x - 12 is shown in the image attached below.

The roots of the equation are (2, 0) and (6, 0).

In Mathematics and Geometry, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. Based on the given quadratic function, we can logically deduce that the graph would be a downward parabola because the coefficient of x² is negative and the value of "a" is less than zero (0).

Since the leading coefficient (value of a) in the given quadratic function y = -x² + 8x - 12 is negative 1, we can logically deduce that the parabola would open downward and the x-intercept (roots) is given by the ordered pair (2, 0) and (6, 0).

In conclusion, the turning point and vertex is given by the ordered pair (4, 4).

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The best-fit linear regression equation gives us the equation y = 0.0804x + 1.1794, where the slope is 0.0804 and the y-intercept is 1.1794.

The trendline equation will give us the equation for the linear regression line, which can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.

Once we have the trendline equation, we can use it to make predictions about future enrollment percentages based on the number of years after 1820.

The slope of the line tells us the rate at which the enrollment percentage is increasing or decreasing over time, and the y-intercept tells us the enrollment percentage when x is equal to zero (i.e., in 1820).

Here we have find that the best-fit linear regression equation gives us the equation

=> y = 0.0804x + 1.1794,

where the slope is 0.0804 and the y-intercept is 1.1794.

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Answer:

Step-by-step explanation:

Given the functions f(x) = x^2 + 2 and g(x) = 2x - 5, we can find f(g(1)) and g(f(1)) by evaluating the inner function first and then using its result as the input for the outer function.

First, let’s find f(g(1)). We start by evaluating the inner function g(1):

g(1) = 2 * 1 - 5 = -3

Now we can use this result as the input for the outer function f(-3):

f(g(1)) = f(-3) = (-3)^2 + 2 = 9 + 2 = 11

Next, let’s find g(f(1)). We start by evaluating the inner function f(1):

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

Now we can use this result as the input for the outer function g(3):

g(f(1)) = g(3) = 2 * 3 - 5 = 6 - 5 = 1

So, f(g(1)) = 11 and g(f(1)) = 1.

Answer:

Step-by-step explanation:

To compare these two values, we first need to convert the mixed number 7 3/8 to an improper fraction. To do so, we multiply the whole number (7) by the denominator of the fraction (8), then add the numerator (3), and put the result over the denominator:

7 3/8 = (7 x 8 + 3) / 8 = 59/8

Now we can rewrite the inequality as:

(59/8) × (4/5) < 59/8

To simplify the left-hand side of the inequality, we multiply the numerators and denominators:

(59/8) × (4/5) = (59 × 4) / (8 × 5) = 236/40 = 59/10

So the inequality becomes:

59/10 < 59/8

To compare these fractions, we need to find a common denominator. The least common multiple of 8 and 10 is 40, so we can convert both fractions to have a denominator of 40:

59/10 = (59 x 4) / (10 x 4) = 236/40

59/8 = (59 x 5) / (8 x 5) = 295/40

Now we can see that 236/40 < 295/40, which means that:

59/10 < 59/8

Therefore, the inequality 7 3/8 × 4/5 < 7 3/8 is true.

The error is in step 2 based on the given statement.

The given statement is[tex]cos(2x) = 1-2sin^2(x),[/tex] and it is not appropriate to simply let 2x = 0.

By doing this, you are only considering a specific case where x = 0, rather than proving the statement for all values of x.

To properly prove the given statement, you should consider using trigonometric identities or double-angle formulas to establish the equivalence between cos(2x) and 1-2sin²(x) for all x.

Thus, the error is in Step 2

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Based on an exponential decay factor, Michael's stereo equipment will be worth $2,116 after 8 years, given the decreasing rate of 9% per year.

The decay factor is represented by (1 - r), where r is the constant periodic decreasing rate.

The decay factor is used in exponential decay functions to determine the depreciated value of an asset.

Current price of stereo equipment = $2,500

Annual decreasing rate = 9%

Decrease factor = 0.91 (100% - 9%)

The number of years = 8 years

Value of the equipment after 8 years = ($2,500 x 0.91^8)

= $2,116.125 ($2,500 x 0.47025)

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Step-by-step explanation:

1-2/6-1/6= 3/6

= 1/2

Marla has led 1/2 of the meetings

Answer: 3.6 is 5% of 72

Step-by-step explanation: If 3.6 is 5% then just multiply 3.6 by 95 to get 68.4 and add to get 72, Yw.

(9,21) is a solution of y=x+12

When is an ordered pair a solution?

To find if an ordered pair (x,y) is a solution to an equation, substitute the x-value and y-value from the ordered pair into the equation, evaluate both sides of the equation individually, and see if the equation is true.

Warning: A common mistake is to use the first coordinate (the x-coordinate, on the left of the ordered pair) as the value on the left side of the equation, and the second coordinate (the y-coordinate, on the right of the ordered pair), as the value for the right side of the equation.  Make sure to substitute the x-value for the "x" in the equation, and the y-value for the "y" in the equation.

Going through the choices:

Point A (24,12)

y = x+12

(12)  ??  (24) + 12

12  ≠  36

not equal -- not a solution

Point B (-12,-24)

y = x+12

(-24)  ??  (-12) + 12

-24  ≠  0

not equal -- not a solution

Point C (9,21)

y = x+12

(21)  ??  (9) + 12

21  =  21

equal -- (9,21) is a solution

Point D (0,-12)

y = x+12

(-12)  ??  (0) + 12

-12  ≠  12

not equal -- not a solution

Answer:

36°

Step-by-step explanation:

x+27+27+90=180

x=36°

Answer:

A = 5

B = 4

C = 8

D = 3

Step-by-step explanation:

This is more of a visual calculation so I can't really explain much about it other than that you match up the side in the prism with the side in the net

Answer: 108

Step-by-step explanation:

If you turn the shape, where the pointy side of the ramp is facing you and you flipped it open and laid out the sides,  that's the "net" image on right

A =  5   the long side of that triangle is also the short side of that rectangle

B = 3    it's the short side of the triangle

C = 8    long part of the rectangle

D =  4  long side of triangle

To find surface area.  Find the area of each of the shapes and add it all up

Top Rectangle:

A=Lw=C*A = 5*8 = 40

Middle Rectangle:

A=Lw = B*C =  3*8 = 24

Bottom Rectangle:

A=LW = 4*C = 4*8=32

Triangles are same

A=1/2 bh = 1/2 D*B = 1/2 * 4* 3 =6

But there are 2 of them so A=12

Now add all the shapes together

A(total)=40+24+32+12=108

The value of x for the triangle is,

x = 58.67 degree

We have to given that;

By Use the Law of Cosines. Find x to the nearest tenth.

And, A triangle ABC is shown in figure.

Hence, We can formulate;

c = √a² + b² - 2ab cos x

16 = √15² + 30² - 2 × 15 × 30 cos x

16 =  √225 + 900 - 900 cos x

256 = 725 - 900 cos x

900 cos x = 725 - 256

900 cos x = 469

cos x = 469 / 900

cos x = 0.52

x = 58.67 degree

Thus, The value of x for the triangle is,

x = 58.67 degree

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The probability that Brent draws mismatched socks is approximately 31/153.

After adding 2 identical black socks, Brent has a total of 18 socks in his drawer. The probability of drawing a white sock on the first draw is 16/18 (since there are 16 white socks out of 18 total socks).

After drawing a white sock on the first draw, there are 15 white socks and 2 black socks left in the drawer. The probability of drawing a black sock on the second draw is 2/17 (since there are 2 black socks out of 17 total socks left). So the probability of drawing a white sock followed by a black sock is;

(16/18) × (2/17) = 32/306

Similarly, the probability of drawing a black sock followed by a white sock is;

(2/18) × (15/17) = 30/306

However, the probability of drawing mismatched socks is;

32/306 + 30/306 = 62/306

= 31/153

Therefore, the probability will be 31/153.

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Answer:

S = 2π(10^2) + 2π(10)(3)

= (200 + 60)π = 260π cm^2

So 250 cm^2 is not a reasonable answer for the surface area of this cylinder.