Q1 - Similar shapes and area
Work out the missing area in each pair of similar shapes.

Answer:

The amount left with her si 62.25

The required value of the production at which the company attained the same profit is 4 thousand skin care items.

Given that,

A certain skincare company's profit in millions of dollars, P(t), can be modeled by the polynomial function P(t) = –2t³ + 8t² + 2t, where t represents the number of skincare items produced, in thousands. Today, the company produces 1 thousand products for a profit of $8 million.

Here,
Plotting the graph of the polynomial equation,

P(t) = –2t³ + 8t² + 2t,
From the graph at x = 4,
The polynomial gives the same value as it gave at x = 1,

Thus, the required value of the production at which the company attained the same profit is 4 thousand skin care items.

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The indefinite integral of given integral ∫√(9+49x^2)dx is ∫√(9+49x^2)dx = [tex]\int\sqrt{(9+49x^2)}dx = 1/98[98x\sqrt{9+49x^2}+9In|7x+\sqrt{9+49x^2}}|]+C[/tex].

In the given question we have to find the indefinite integral.

The given integral is

∫√(9+49x^2)dx

Now finding the indefinite integral using the C for the constant of integration.

An integral is regarded to be indefinite if it has no lower or upper bounds. In mathematics, the most generic antiderivative of f(x) is known as an indefinite integral and expressed by the expression f(x) dx = F(x) + C.

We can write it as:

[tex]\int\sqrt{(9+49x^2)}dx = \int\sqrt{[(3)^2+(7x)^2]}dx[/tex]

Taking common 7^2

[tex]\int\sqrt{(9+49x^2)}dx = 7\int\sqrt{[(3/7)^2+(x)^2]}dx[/tex]

Using the formula

[tex]\int\sqrt{(a^2+x^2)}dx = \frac{1}{2}\times\sqrt{a^2+x^2} + \frac{1}{2}\cdot a^2In|x+\sqrt{a^2+x^2}|+C[/tex]

[tex]\int\sqrt{(9+49x^2)}dx = 7[x\sqrt{(3/7)^2+x^2}+\frac{1}{2}\cdot(\frac{3}{7})^2In|x+\sqrt{(3/7)^2+x^2}|]+C[/tex]

[tex]\int\sqrt{(9+49x^2)}dx = 7[x\sqrt{9/49+x^2}+1/2\cdot(9/49)In|x+\sqrt{(9/49)+x^2}|]+C[/tex]

[tex]\int\sqrt{(9+49x^2)}dx = 7[x\cdot\frac{\sqrt{9+49x^2}}{7}+1/2\cdot(9/49)In|x+\frac{\sqrt{9+49x^2}}{7}|]+C[/tex]

[tex]\int\sqrt{(9+49x^2)}dx = 7[x\cdot\frac{\sqrt{9+49x^2}}{7}+(9/98)In|\frac{7x+\sqrt{9+49x^2}}{7}|]+C[/tex]

[tex]\int\sqrt{(9+49x^2)}dx = 1/98[98x\sqrt{9+49x^2}+9In|7x+\sqrt{9+49x^2}}|]+C[/tex]

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The ordered pair of the inequality 6x + 5y < -15 are (2, 0) and (3, 2).

When two expressions are connected by a sign like "not equal to," "greater than," or "less than," it is said to be inequitable. The inequality shows the greater than and less than relationship between variables and the numbers.

Considering the provided inequality and the provided ordered pairs, we will check it by replacing one by one in the inequality; if the inequality is fulfilled, they are the solution; otherwise, they are not, so we obtain:

(2, 0):

6x + 5y > 2

6(2)+5(0)>2

12+0>2

12>0

Since the inequality holds, then the ordered pair is a solution.

(7, -8):

6x + 5y > 2

6(7)+5(-8) > 2

42-40>2

2>2

Since the inequality is not satisfied, then the ordered pair is not a solution.

(3, 2):

6x + 5y > 2

6(3)+5(2) >2

18+10>2

28>2

Since the inequality holds, then the ordered pair is a solution.

(-8, 6):

6x + 5y > 2

6(-8)+5(6) >2

-48 + 30 > 2

-18 > 2

Since the inequality is not satisfied, then the ordered pair is not a solution.

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Answer:Find the Inverse f (x)=-2x+9Find the Inverse f (x)=-2x+9 f (x) = −2x + 9 f (x) = - 2 x + 9 Write f (x) = −2x+ 9 f (x) = - 2 x + 9 as an equation. y = −2x+9 y = - 2 x + 9 f (x) = −2x + 9 f (x) = - 2 x + 9 Write f (x) = −2x+ 9 f (x) = - 2 x + 9 as an equation. y = −2x+9 y = - 2 x + 9

Step-by-step explanation:

Answer:

[tex] \huge{ \boxed{v > 5.6}}[/tex]

Step-by-step explanation:

[tex]0.5v + 0.03 >2.8[/tex]

In order to solve this inequality, first subtract 0.03 from both sides of the inequality to isolate 0.5v

[tex]0.5v + 0.03 - 0.03 > 2.83 - 0.03 \\ 0.5v > 2.8[/tex]

Next divide both sides by 0.5

[tex] \frac{0.5v}{0.5} > \frac{2.8}{0.5} \\ v > 5.6[/tex]

We have the final answer as

[tex] \bold{v > 5.6}[/tex]

The tension in the support cable is 2731 N

Given that:

A cable running from the top of a telephone pole creates a horizontal pull of 889 Newtons (N) and a support cable running to the ground is inclined 71° from the horizontal.

This can sketch as shown in the attached image. Considering the image:

T is the  tension in the support cable

Using trigonometric ratio:

cos 71° = 889/T    (adjacent/hypotenuse)

T = 889/cos 71°

T = 2731 N

Therefore, the tension in the support cable is 2731 N

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Answer:

Step-by-step explanation:

The value of x inside the triangle is 24°

How to calculate the value of x(angle)?

The angle of the straight line = 180°

156° + x = 180°

x = 180° - 156°

x = 24°

Hence, the value of x inside the triangle is 24°.

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The correct answer is option B which is  -1.67 is the expected value.

It is a branch of mathematics that deals with the occurrence of a random event.

You have a higher probability of losing money than gaining it due to the distribution of probability.  

There are more chances to lose than win.

As a result, you are expected to lose money.

Only -1.67 is the only negative option, which means that we don't even need to do real math since there's already only one choice.

Hence, the correct answer is option B which is  -1.67 is the answer.

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Answer:

You will Gain 5 points.

Step-by-step explanation:

Got it right on the test!

Answer:

Step-by-step explanation:

If -8 is less than or equal to w, and 3 is greater than or equal to w, then we can combine these two inequalities into a single inequality by combining the left and right sides:

-8 ≤ w ≤ 3

This inequality can be read as "w is greater than or equal to -8 and less than or equal to 3." It represents all values of w that are greater than or equal to -8 and less than or equal to 3, including -8 and 3.

I hope this helps! Let me know if you have any questions.

Answer:

Below

Step-by-step explanation:

For REAL numbers, the discriminant of the Quadratic Formula has to be a positive value (or zero)

b^2 - 4ac   >= 0         where   a = m       b = m-2        c = -m-1  

(m-2)^2 - 4(m)(-m-1) >=0  

5m^2 +4   >= 0      

       5 m^2 +4     <===== will always be >= 0   for any value of 'm'

                                     so all roots will be real

An inequality that represents the height h, in inches, of a person allowed to ride rollar coaster is h ≥  48

If the equation or inequality contains variable terms, then there might be some values of those variables for which that equation or inequality might be true. Such values are called solution to that equation or inequality. Set of such values is called solution set to the considered equation or inequality.

Let h represent for the height of any member allowed to ride rollar coaster

From the question we know that everyone's height must be greater that 48 inches.

It means the inequalities would be

h ≥  48 tall inches

Thus, an inequality that represents the height h is h ≥  48

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Answer:

Not parallel.

Step-by-step explanation:

Given:

Linear Pair

Two adjacent angles that sum to 180° (two angles which when combined together form a straight line).

Angles 9 and 10 form a linear pair:

⇒  m∠9 + m∠10 = 180°

⇒  110° + m∠10 = 180°

⇒  m∠10 = 70°

Angles 7 and 8 form a linear pair:

⇒  m∠7 + m∠8 = 180°

⇒  m∠7 + 70° = 180°

⇒  m∠7 = 110°

Vertical Angles Theorem

When two straight lines intersect, the opposite vertical angles are congruent.

Corresponding Angles Postulate

When a straight line intersects two parallel straight lines, the resulting corresponding angles are congruent.

As line p is parallel to line q, and according to the vertical angles theorem and the corresponding angles postulate:

If line a was parallel to line b, then according to the corresponding angles postulate:

As m∠1 ≠ m∠8 and m∠2 ≠ m∠7 then lines a and b are not parallel.

Answer:

x = 1

y = -1

Step-by-step explanation:

6x + 3y = 3 -------------(1)

3x - y = 4 ----------------(2)

Multiply Eq. (2) by 2

2(3x - y) = 4 × 2

6x - 2y = 8 -------------(3)

Subtract Eq. (3) from Eq. (1)

6x + 3y - (6x - 2y) = 3 - 8

6x + 3y - 6x + 2y = -5

3y + 2y = -5

5y = -5

Divide 5 to both sides

y = -5/5

Put y = -1 in Eq. (1)

6x + 3(-1) = 3

6x - 3 = 3

Add 3 to both sides

6x = 3 + 3

6x = 6

Divide 6 to both sides

[tex]\rule[225]{225}{2}[/tex]

The proof is shown in the image.

Reasons for statement

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The area of the given figure composed of a rectangle and two semicircles is 68.56 unit².

The space around a shape's perimeter or boundary is known as its area. To determine the areas of various forms, many mathematical formulae are available.

Here we have

composed of a rectangle and two semicircles.

Dimensions of the rectangle are,

Length = 14 and breadth = 4  

=>Area of the rectangle = 14 × 4 = 56 units²

If we join the two semicircles,

we will have one circle with a diameter of 4 units

=> As we know Radius of the circle = Diameter/2

=> Radius of circle = 4/2 = 2 units

=> Area of the circle = 2πr

= [tex]2 \times\frac{22}{7} \times 2[/tex] = 12.56 unit²  

Area of the figure = Area of rectangle + Area of the circle

= 56 units² + 12.56 unit²  

= 68.56 unit²

Therefore,

The area of the given figure composed of a rectangle and two semicircles is 68.56 unit².

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Using the normal distribution, it is found that a student has to score 0.675 standard deviations above the mean to be publicly recognized.

In a normal distribution with mean  and standard deviation , the z-score of a measure X is given by:

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Negative 2.5 x greater-than-or-equal-to negative 30 is the equivalent expression of 50 minus 2.5 x greater than or equal to 20

Equivalent expressions can be defined as expressions that are the same, even though they may look a little different. If you plug in the same variable value into equivalent expressions, they will give you the same value when you simplify

Given: 50 minus 2.5 x greater than or equal to 20

50 - 2.5x ≥ 20

Subtract 50 from both sides:

-2.5x ≥ 20-50

-2.5x ≥  -30

Therefore, the equivalent expression is "negative 2.5 x greater-than-or-equal-to negative 30"

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Answer:

A = 25t + 300

B = 55t + 150

Less than 5 hours, company B would be cheaper.  At 5 hours the cost is equal.  After 5 hours, company A would be cheaper.

Step-by-step explanation:

Company B would be cheaper for lower hours.

The the two equation equal to each other and solve for to find out when the costs will be equal.

25t + 300 = 55t + 150  Subtract 25t from both sides

300 = 30t + 150  Subtract 150 from both sides

150 = 30t  Divide both sides by 30

5 = t

At 5 hours the cost are the same, after 5 hours company A would be cheaper.

Answer:

1:4

Step-by-step explanation:

4:16 = 1:?

4/4 = 1

16/4 = 4

= 1:4

The distance the pear will be in the air if his Dad sees him and catches if at 6 feet above the ground is 19/30 meters.

The term distance can be defined as a numerical qualitative measurement of how far apart  points are from each other.

Distance is defined by the formula

d=v*t

The given parameters are:

v=30

s=6

height=25

Applying these parameters into the formula

d=30*t

d=30t

If the Dad stands 6 feet above the ground, we have

[tex]\frac{25-6}{30}[/tex]

=[tex]\frac{19}{30}[/tex]

Therefore the pear will be 19/30 feet in the air if the Dad sees him and catches  it at 6 feet above the ground.

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1) h=1/5 k

(10/2 = 1/5, 20/4 = 1/5... etc)

2) k=5/4 h

(5/4 = 5/4, 10/8=5/4... etc)

3) h=5/1 k

(25/5 = 5/1, 30/6 = 5/1 ... etc)

Hope this helps!

:]

By using the concept of probability, it can be calculated that-

1)   P(Ace [tex]\cup[/tex] 5) = [tex]\frac{8}{52}[/tex]

2)  P(Ace [tex]\cap[/tex] 5) = 0

3)  P([tex]Even^c[/tex]) = [tex]\frac{20}{52}[/tex]

4)   P(Even [tex]\cup[/tex] Black) = [tex]\frac{36}{52}[/tex]

5)   P(Face [tex]\cap[/tex] Diamond) = [tex]\frac{3}{52}[/tex]

What is probability?

Probability gives us the information about how likely an event is going to occur.

Probability is calculated by Number of favourable outcomes divided by the total number of outcomes.

Probability of any event is greater than or equal to zero and less than or equal to 1.

Probability of sure event is 1 and probability of unsure event is 0.

1) Number of ace card = 4

  Number of '5' card = 4

  P(Ace [tex]\cup[/tex] 5) = [tex]\frac{4}{52} + \frac{4}{52}[/tex]

                    = [tex]\frac{8}{52}[/tex]

2) P(Ace [tex]\cap[/tex] 5) = 0

3) Number of odd numbered cards = [tex]5 \times 4[/tex] = 20 (Assuming the ace is odd)

 Probability of getting an odd numbered card = [tex]\frac{20}{52}[/tex]

  P([tex]Even^c[/tex]) = [tex]\frac{20}{52}[/tex]

4) Number of black coloured even numbered card = 10

   Number of even numbered card = 20

   Number of card of black colour = 26

  P(Even [tex]\cup[/tex] Black) = [tex]\frac{20}{52} + \frac{26}{52} - \frac{10}{52}[/tex]

                              = [tex]\frac{36}{52}[/tex]

5) Number of face cards of diamond = 3

   P(Face [tex]\cap[/tex] Diamond) = [tex]\frac{3}{52}[/tex]

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The range of u(x) = {-15, -5,1,3,1, -5, -15} and range of v(x)= {-3,-2, -1,0,1,2,3} for the domain of x = { -3,-2,-1,0,1,2,3}

The domain of a function is defined by the set of the values of independent variables and the ranges are the set of values of dependent variables.

The domains act as an input of function where ranges act as an output.

We select domain randomly as domain are the set of independent variables. But how will be the ranges must depend on the function given us.

we are given 2 function u(x) = -2x²+3 and v(x)

for the 1st function the set of domain and ranges are different, but the domain and ranges have the same values for the 2nd function.

we select domain x= {-3, -2, -1, 0, 1, 2, 3}

the ranges for u(x) ={ -15, -5, 1, 3, 1, -5 , -15}

because, u(-3) = -2(-3)²+3 = -15

                u(-2) = -2(-2)²+3 =-5

               u(0) = 3

similarly, for the same set of x value, v(-3) = -3

                                                         v(-2) = -2

                                                         v(0) =0

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The leading team has scored 0.13 points per game for the season that the second place player.

What is PPG Statistics?

First, consider the PPG of the current leader of the season from among the two basketball players.

The total number of points obtained by the current leader is equal to 2112.

The total number of games played by the current leader is equal to 77.

The points per game obtained by the current leader can be calculated as follows,

PPG = Total Points Obtained / Total Number of Games

[tex]\implies PPG=\frac{2112}{77}\\\implies PPG=27.42[/tex]

Now, consider the PPG of the second place player for the season.

The total number of points obtained by the second place player is equal to 2020.

The total number of games played by the second place player is equal to 74.

The points per game obtained by the second place player can be calculated as follows,

PPG = Total Points Obtained / Total Number of Games

[tex]\implies PPG=\frac{2020}{74}\\\implies PPG=27.29[/tex]

Subtracting the points per game of both players, we get

[tex]27.42-27.29=0.13[/tex]

Hence, the leading team has scored 0.13 points per game (PPG) for the season compared to the second place player.

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Answer:

  75°

Step-by-step explanation:

You want to know the measure of the image angle B'C'A' after angle A'B'C' is dilated by a factor of 4 about point P. Angle BCA is 75°.

Dilation does not affect angle measures, so angle B'C'A' has the same measure as angle BCA, 75°.

  m∠B'C'A' = 75°

The simplification of the given equation is -630y² + 81y + (+2.6y) * 90y² = 0.

What is the domain of the function?

The range of values that we are permitted to enter into our function is known as the domain of a function. The x values for a function like f make up this set (x). A function's range is the collection of values it can take as input. After we enter an x value, the function outputs this sequence of values.

We have,

1.6y+y-4/15y+7/6y=21/3

Move all terms to the left:

1.6y+y-4/15y+7/6y-(21/3)=0

Domain of the equation: 15y!=0

y!=0/15

y!=0

y∈R

Domain of the equation: 6y!=0

y!=0/6

y!=0

y∈R

Add all the numbers together, and all the variables

1.6y+y-4/15y+7/6y-7=0

Add all the numbers together, and all the variables

2.6y-4/15y+7/6y-7=0

calculate fractions

2.6y+(-24y)/90y²+105y/90y²-7=0

multiply all the terms by the denominator

(2.6y) * 90y² + (-24y) + 105y - 7 * 90y² = 0

Add all the numbers together, and all the variables

(2.6y) * 90y² + (-24y) + 105y - 7 * 90y² = 0

Add all the numbers together, and all the variables

105y+(+2.6y)*90y² + (-24y) - 7*90y² = 0

multiply elements

-630y+105y+(+2.6y)*90y² + (-24y) = 0

get rid of parentheses

-630y² + 105y + (+2.6y)*90y² - 24y = 0

Add all the numbers together, and all the variables

-630y² + 81y + (+2.6y) * 90y² = 0

Hence, the simplification of the given equation is -630y² + 81y + (+2.6y) * 90y² = 0.

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