i am supposed to find the volume of this pyramid

The Solution:

The given inequality is

Clearing the brackets, we get

Collecting the like terms, we get

So, the solution is true for all real values of x.

The interval notation of the solution is

Answer:

3

Step-by-step explanation:

Given expression:

[tex]\dfrac{(14^2-13^2)^{\frac{2}{3}}}{(15^2-12^2)^{\frac{1}{4}}}[/tex]

Following the order of operations, carry out the operations inside the parentheses first.

Apply the Difference of Two Square formula  [tex]x^2-y^2=\left(x+y\right)\left(x-y\right)[/tex]

to the operations inside the parentheses in both the numerator and denominator:

[tex]\implies \dfrac{((14+13)(14-13))^{\frac{2}{3}}}{((15+12)(15-12))^{\frac{1}{4}}}[/tex]

Carry out the operations inside the parentheses:

[tex]\implies \dfrac{((27)(1))^{\frac{2}{3}}}{((27)(3))^{\frac{1}{4}}}[/tex]

[tex]\implies \dfrac{(27)^{\frac{2}{3}}}{(81)^{\frac{1}{4}}}[/tex]

Carry out the prime factorization of 27 and 81.

Therefore, rewrite 27 as 3³ and 81 as 3⁴:

[tex]\implies \dfrac{(3^3)^{\frac{2}{3}}}{(3^4)^{\frac{1}{4}}}[/tex]

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

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

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

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

[tex]\implies 3^{(2-1)}[/tex]

[tex]\implies 3^1[/tex]

[tex]\implies 3[/tex]

Given that,

→ ((14² - 13²)^⅔)/((15² - 12²)^¼)

Evaluating the problem,

→ ((14² - 13²)^⅔)/((15² - 12²)^¼)

→ ((196 - 169)^⅔)/((225 - 144)^¼)

→ (27^⅔)/(81^¼)

→ ((3³)^⅔)/((3⁴)^¼)

→ (3²)/3

→ 9/3 = 3

Therefore, the solution is 3.

It is expected that the cost of gas will increase as the number of gallons increases and decrease as the number of gallons decrease. This is a direct proportion

As such, if

12 gallons of gas cost $26.68

15 gallons will cost 15/12 * $26.68

= $33.35

15 gallons of gas will cost $33.35

Let's recall one of the properties of exponents:

Therefore, in our exercise we have:

1. If the cake rises by ¹/₃ as it bakes, the number of cups of cake batter needed for the four cakes is 140.

2. If ¹/₄ in. is used between layers and ¹/₂ in. is used on the top and sides, the number of cups of icing needed for the four cakes is 47.

The number of cups of cake batter and icing can be determined using the mathematical operations of multiplication, addition, division, and subtraction.

First, the volumes of each cake and its batter are calculated using the given dimensions and the rise.

Using the division operation, the number of cups of cake batter for each cake is determined and multiplied by four.

We understand that the normal volume of the cake will increase with the icing, helping us to calculate the increased volume after the icing.

The difference between the two volumes becomes the volume of the icing required, which is divided by 14.4 in³ to get the number of cups of icing required.

The volume of each cake = Length x Width x Height

Length = 14 inches

Width = 12 inches

Height = 4 inches (2 x 2)

= 12 x 14 x 4

= 672 in³.

Rise of the cake as it bakes =  ¹/₃

The normal volume before rising = 1

Risen volume = 1¹/₃

1¹/₃ = 672 in³

The normal volume of cake batter before the ¹/₃ rise = 504 in³ (672/1¹/₃).

1 cup = 14.4 in³, the total cups for each cake = 35 cups (504 in³/14.4 in³).

The total cups of cake batter for the 4 cakes = 140 cups (35 x 4).

The total quantity of icing = 1¹/₄ (¹/₄ + ¹/₂ + ¹/₂).

The new volume after the icing = 840 in³ (672 x 1¹/₄)

The difference in volume after the icing = 168 in³ in (840 in³ - 672 in³)

If 1 cup = 14.4 in³, the cups of icing for each cake = 11.67 cups (168 in³/14.4 in³).

The total cups of icing for the 4 cakes = 47 cups (11.67 x 4).

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Four double-layered cakes, each 12.0 in. x 14.0 in., have been ordered for a special event.  Each layer is 2.0 in. high.

a) If the cake rises by ¹/₃ as it bakes, how many cups of cake batter are needed? (1 cup = 14.4 in³.  Hint: The ¹/₃ rise should be treated as a constant.)

b) How many cups of icing are needed if ¹/₄ in. is used between layers and ¹/₂ in. is used on the top and sides? (Assume icing is not layered on top of the icing. 1 cup = 14.4 in³.)

Solution:

For this case we know that the width is given by:

w = 6x +8

The lenght is given by:

l= 12x +16

And the perimeter would be given by:

P= 2l +2w = 2(12x+16)+ 2(6x+8)= 24x+32 +12x+16=36x + 48

And then the ratio would be:

we have that

the volume of the cube is equal to

V=b^3

64=b^3

b^3=4^3

b=4 unit

see the attached figure

the surface area of the new figure is equal to

SA=2B+PH

where

B is the area of the base

P is the perimeter of the base

H is the height

we have

B=4*8=32 unit2

P=2(4+8)=24 unit

H=2 unit

so

SA=2(32)+24*2

SA=64+48

SA=112 unit2

We will have the following:

*Initially, as x increases, y decreases.

*The slope of the graph is equal to -1 for all x between x = 0 & x = 3.

*Starting at x = 3, the function value y increases as x increases.

*The slope of the graph is equal to 3 for x between x = 3 & x = 5.

*For x between x = 0 & x = 4, the function value y ≤ 0.

*For x between x = 4 & x = 8, the function value y ≥ 0.

Answer

The two coordinates on the line include

(0, -1.5) and (-4.5, 0)

The graph of the line is presented below

Explanation

We are asked to plot the grap of the given equation of a straight line.

To do that, we will obtainthe coordinates of two points on the line.

These two points will preferrably be the intercepts of the line.

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

when x = 0

y = 0 - (3/2)

y = -(3/2)

y = -1.5

First coordinate and first point on the line is (0, -1.5)

when y = 0

0 = (-x/3) - (3/2)

(x/3) = -(3/2)

x = (-3) (3/2)

x = -(9/2)

x = -4.5

Second coordinate and second point on the line is thus (-4.5, 0)

So, to plot the line, we just mark these two points and connect them to each other.

The graph of this line is presented under 'Answer' above.

Hope this Helps!!!

Exponential Decay

The model for the exponential decay of a quantity Mo is:

Where λ is a constant and t is the time.

The half-life of a radioactive isotope is the time it takes to halve its initial mass. It can be calculated by making M = Mo/2 and solving for t:

It's required to calculate the remaining mass of an isotope of Mo = 175 gr after 5 half-lives have passed, that is. we must calculate M when t is five times the value calculated above.

Substituting in the model:

Simplifying (the value of λ cancels out):

Rounding to the nearest gram, 5 grams of the radioactive isotope will be left after the required time.

Answer:

x = 34 degrees, y = 73

Step-by-step explanation:

Since the triangle is isosceles, the base angles are congruent (equal). First, find the supplement angle by doing 180-107, which gives you 73 for the base angles, which include y. Now there is a theorem that states the 2 remote interior angles are equivalent to the exterior angle, which means 107 = 73 + x. This gives us x = 34

I hope this helps!

Use the same process for the second one.

We have to find the value of z and x.

We assume that lines g and h are parallel.

Then, z and the angle with measure 85° are consecutive interior angles.

As they are conscutive interior angles, their measures add 180°.

Then, we can write:

Then, we can relate the angle with measure z with the angle with measure (6x-109). They are vertical angles and, therefore, they have the same measure.

Then, we can write:

Answer: z = 95 and x = 34.

Notice that the x coordinate of both points is the same, therefore the line is a vertical line.

Answer: The slope is undefined.

FOR THE PAIRS TO BE OPPOSITE IT MEANS THE NUMBERS SHOULD ALSO CONTRAST IN SIGNS.

47 AND -47 ARE OPPOSITE

74 AND - 74 ARE OPPOSITE

|4|=4 AND -4 ARR OPPOSITE

47 AND |-47|=47 ARE NOT OPPOSITE BECAUSE THEY BOTH HAVE THE SAME SIGNS.

THE LAST OPTION IS THE ANSWER.

If the price changes from $105,300 to $104,399, It means that there is a decrease in price.

Decrease = 105,300 - 104,399 = $901

The percentage decrease is gotten by dividing the decrease by the initial price and multiplying by 100. It becomes

By rounding up to the nearest whole number, it becomes 1%

The percent decrease is 1%

17 Liters

Step 1

find the total area to paint

we need to assume the floor wont be painted, so the total are to paint is

the are of a rectangle is gieven by:

so, the total area will be

so , the area to paint is 680 square meters

Step 2

finally, to know the number of Liters need , divide the amount ( total area) by the rate of the paitn, so

so, the total paint needes is 17 Liters, and paint comes in 1-L cans, so

therefore, the answer is

17 Liters

I hope this helps you

We have the next points (1,2) and (3,-6)

The slope-intercept form is an equation as follows:

Then, we need to change the original equation in this equivalent:

Dividing the total equation by 7, we have:

Therefore, the slope-intercept form is:

The values of f(0), f(2) and f(-2) for the polynomial f(x) = [tex]-x^{3} +7x^{2} -2x+12[/tex] are 12, 28 and 52 respectively.

According to the question,

We have the following information:

f(x) = [tex]-x^{3} +7x^{2} -2x+12[/tex]

Now, to find the value of f(0), we will put 0 in place of x.

f(0) = [tex]-0^{3} +7(0)^{2} -2(0)+12[/tex]

f(0) = 0+7*0-0+12

(When a number has some power then it means that in order to solve this we have expand the expression and multiply the number as many times as the power is given. For example, in the case of 3 as power, we will multiply any number 3 times and in case of 2 as power, we will multiply the given number 2 times.)

f(0) = 0+0-0+12

f(0) = 12

Now, to find the value of f(2), we will put 1 in place of x:

f(2) = [tex]-2^{3} +7(2)^{2} -2(2)+12[/tex]

f(2) = -8+7*4-4+12

f(2) = -8+28-4+12

f(2) = 40 -12

f(2) = 28

Now, to find the value of f(2), we will put -2 in place of x:

f(-2) = [tex]-(-2)^{3} +7(-2)^{2} -2(-2)+12[/tex]

f(-2) = -(-8) + 7*4+4+12

f(-2) = 8+28+4+12

f(-2) = 52

Hence, the value of f(0) is 12, f(2) is 28 and f(-2) is 52.

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S = 2(a*b + a*c + b*c)

= 2 (12*15 + 12*6 + 15*6)

= 2 (342)

= 684 ft^2

Probability that the student got a 'C' GIVEN they are female = number of females that got a C in the test/number of females

From the information given,

number of females that got a C in the test = 12

number of females = 30

Thus,

Probability that the student got a 'C' GIVEN they are female = 12/30

We would simplify the fraction by dividing the numerator and denominator by 6. Thus,

Probability that the student got a 'C' GIVEN they are female = 2/5

The number of miles that she need to bike on the last day so that her average (mean) is 63 miles per day is 62 miles.

The mean is the average of a set of numbers. Let the biking of the last day be represented as x. This will be:

(61 + 67 + 66 + x) / 4 = 64

(194 + x) / 4 = 64

Cross Multiply

194 + x = 64 × 4

194 + x = 256

Collect like terms

x = 256 - 194

x = 62

The miles is 62 miles.

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The profit on one standard DVD player is $59 and on one portable DVD player is $69.

If there are s number of standard DVD player then total profit on standard DVD players is $59s. Simillarly total profit on portable DVD players is $69p.

The total profit on DVD player shoul be at least $345, which means total profit on DVD players is $345 or more than $345.

The linear inequalty for total profit is,

The graph of the linear inequality is,

In graph, lines pointing away the origin represent the region for the equation.

The factors are -3 and -1

What is a Quadratic equation ?

A second-degree equation of the form ax² + bx + c = 0 is known as a quadratic equation in mathematics. Here, x is the variable, c is the constant term, and a and b are the coefficients. Since x is a second-degree variable, this quadratic equation has two roots, or solutions.

The given expression is,

2x² + 8x + 6

Put it equal to 0 so that we can solve for 'x'

2x² + 8x + 6 = 0

Now, its factors are 6x and 2x

2x² + 6x + 2x + 6 = 0

2x(x + 3) + 2(x + 3) = 0

To cross check your solution is correct or not. You've to just see the the brackets value should be same after taking common. Here the bracket value is (x+3) which is same.

(2x + 2) (x+3) = 0

split the values to solve further,

2x + 2 = 0                |         x + 3 = 0

2x = -2                     |          x = -3

x = -2/2

x = -1

Hence, the factors are -3 and -1

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Given the quadratic equation:

Let's solve for the following:

• (a) g(-3)

To solve for g(-3), substitute -3 for x and evaluate.

Thus, we have:

Hence, we have:

g(-3) = 46

• (b) To determine the coordinates of the point given in question (a).

In the function, g(x) can also be written as y.

Thus, from g(-3), we have the following:

x = -3

y = 46

When x = -3, the value of y = 46

In point form, we have the coordinates:

(x, y) ==> (-3, 46)

Therefore, the coordinates of the given point by the answer in (a) is:

(-3, 46)

• (c) Evaluate g(2a).

To evaluate g(2a), substitute 2a for x in the equation and evaluate.

Thus, we have:

ANSWERS:

• (a) g(-3) = 46

• (b) (-3, 46)

• (c) g(2a) = 12a² - 10a + 4

       A vertex is a point on a polygon where two rays or line segments meet, the sides, or the edges of the object come together. Vertex is the plural form of vertices.

f(x)=3(x−1)2

Find the given parabola's characteristics.

Lessen the steps you tap...

Vertex form is to be used, y=a(x−h)2+k,

to calculate the values of a, h, and k.

a=3

h=1

k=0

The parabola widens because the value of an is positive.

opens up

Find the (h,k) vertex. ( 1 , 0 )

Calculate p, the distance between the focus and the vertex.

To continue, tap...

1/ 12

Locate your focus.

To continue, tap... ( 1 , 1/ 12 )

By identifying the line that connects the vertex with the focus, you may determine the axis of symmetry.

x = 1

 The horizontal line that results from deducting p from the vertex's y-coordinate k depends on whether the parabola opens up or down. This line is known as the directrix.

y = k − p

Simplify the formula after substituting the known p and k values.

y = − 1 /12

Analyze and graph the parabola using its characteristics.

Direction: opens up

vertices: ( 1, 0 )

Focus: ( 1 , 1 /12 )

x = 1 is the symmetry axis.

Direction: y = /1 12.

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Solution

Given that:

Option A.

SP = 2a+5

RQ= 5a-1

ST = 3b-3

SQ = 7b-9

RQ=?

TQ=?

SP = RQ

2a+5 = 5a-1

Solve for a

5+1 = 5a-2a

6 = 3a

6/3 = a

2=a

RQ= 5a-1 = 5(2)-1 = 10-1 = 9

RQ= 9

ST + TQ = SQ

ST= TQ

TQ= 3b-3

3b-3+3b-3= 7b-9

Solve for b

6b-6 = 7b-9

-6+9 = 7b-6b

3=b

TQ = 3b-3= 3(3)-3= 9-3 =6

TQ= 6

Answer: You need to spend more than $5.00

Step-by-step explanation: