1 optimization calculus question, 50 pts please help

Answer:

V = 1/3 π r^2 h

Step-by-step explanation:

V = 1/3 area base*height

1. work out the area

2. base*height

3. area*answer

4. you get full answer

The inequality that represents the number of each type of calculator she can buy will be 25x + 65y ≤ 800

Given that teacher needs to purchase some scientific calculators for $25 each and some graphing calculators for $65 each.

Consider that x equal the number of scientific calculators and y equal the number of graphing calculators she buys She can only spend up to $800 for the calculators.

The inequality that represents the situation becomes as;

25x + 65y ≤ 800

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Part 1

E. There is a pair of congruent angles and a pair of congruent right triangles, so the triangles are similar by the AA \(\sim\) postulate.

Part 2

Note that 4 ft 6 in = 4.5 ft. Using the fact that corresponding sides of similar triangles have proportional side lengths,

\(\frac{x}{14}=\frac{4.5}{4} \implies \boxed{x=15.75 \text{ ft}}\)

The number of the hamburgers and sodas sold are respectively; 69 and 9

We are given;

Cost of Hamburgers = $1.50

Cost of Sodas = $0.50

Total amount made from the sales = $108.50

Total number of hamburgers and sodas sold = 78

Now,

Let x represent the number of hamburgers sold

Let y represent the number of sodas sold

Thus, the simultaneous equation to represent cost and numbers sold are;

1.50x + 0.50y = 108.50   -----(eq 1)

x + y = 78    -----(eq 2)

Making x the subject in eq 2 gives;

x = 78 - y    ----(eq 3)

Plug in 178 - y for x in eq 1 to get;

1.50(78 - y) + 0.50y = 108.50

117 - 1.5y + 0.5y = 108.5

y = 8.5 ≈ 9

Thus;

x = 78 - 9

x = 69

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The average speed formula is given by:

\(\displaystyle\sf \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\)

Let's calculate the total distance first. The car travels 120 miles from point A to point B, and then returns the same distance. Therefore, the total distance is \(\displaystyle\sf 2 \times 120 = 240\) miles.

Now, let's calculate the total time. The time taken to travel from A to B can be calculated using the formula \(\displaystyle\sf \text{Time} = \frac{\text{Distance}}{\text{Speed}}\).

For the journey from A to B:

\(\displaystyle\sf \text{Time}_1 = \frac{120}{30} = 4\) hours

For the return journey from B to A:

\(\displaystyle\sf \text{Time}_2 = \frac{120}{40} = 3\) hours

The total time for the round trip is the sum of the individual times:

\(\displaystyle\sf \text{Total Time} = \text{Time}_1 + \text{Time}_2 = 4 + 3 = 7\) hours

Now, we can calculate the average speed:

\(\displaystyle\sf \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{240}{7} \approx 34.29\) miles per hour.

Therefore, the average speed for the round trip is closest to 34.29 miles per hour.

Answer: P=60.16m

Step-by-step explanation:

Answer: y=8 and x=8 radical 3

Answer:

x=403

Step-by-step explanation:

Let's solve your equation 

Hey there! I'm happy to help!

We want to get x all by itself on one side to find out what it is equal to.

We add 5 to both sides.

3x=40

And we divide both sides by 3.

x=13 1/3

Have a wonderful day! :D

Thus, the area of ΔHIJ using the Heron's formula is found as 580.47 square centimeter.

Heron of Alexandria (c. 62 ce) is credited with developing the Heron's formula, which determines the area of a triangle in regards of the lengths of its sides. If the side lengths are represented by the symbols a, b, and c: √s(s - a)(s - b)(s - c)

where s = half the perimeter,

s = (a + b + c)/2.

given data:

In ΔHIJ,

semi -perimeter s = (i + j + h) / 2

s = (33 + 61 + 39) / 2

s = 66.5

Now,

s - h = 66.5 - 33 = 33.5

s - i = 66.5 - 61 = 5.5

s - j = 66.5 - 39 = 27.5

area of ΔHIJ = √s(s - h)(s - i)(s - j)

area of ΔHIJ = √66.5*33.5*5.5*27.5

area of ΔHIJ = √336947.1875

area of ΔHIJ = 580.47

Thus, the area of ΔHIJ using the Heron's formula is found as 580.47 square centimeter.

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Answer:

20 blue marbles

35 green marbles

Step-by-step explanation:

Let b = number of blue marbles.

"there are 15 more green marbles than blue marble"

The number of green marbles is b + 15.

The total number of marbles is

b + b + 15

The total number of marbles is 55.

b + b + 15 = 55

2b = 40

b = 20

Blue marbles: 20

Green marbles:

b + 15 = 20 + 15 = 35

Green marbles: 35

The angle z is in the third quadrant, the value of cosz is negative. Hence, cosz = -3/5.So, the value of cosz is -3/5.

Given sinz = -4/5 for pi < z < (3pi)/2, we need to find the value of cosz. We can use the trigonometric identity of Pythagorean theorem to find the value of cosz.

According to Pythagorean theorem, sin2θ + cos2θ = 1, where θ is the angle in the right-angled triangle and sin, cos are the trigonometric ratios.

The negative sign for the given sinz indicates that the angle z is in the third quadrant. So, we can take the help of the unit circle to find the value of cosz as shown below:

Here, we have used the Pythagorean identity of sin2z + cos2z = 1 on the unit circle to find the value of cosz. Since the value of sinz is already given, we can find the value of sin2z as: sin2z = sinz x sinz = (-4/5) x (-4/5) = 16/25

Then, we can substitute the value of sin2z in the Pythagorean identity as: cos2z = 1 - sin2z = 1 - (16/25) = 9/25We need to find the value of cosz.

So, we can take the square root of cos2z as: cosz = ±(√(9/25)) = ±(3/5)The sign of cosz can be determined by considering the quadrant of the angle z.

Since the angle z is in the third quadrant, the value of cosz is negative. Hence, cosz = -3/5.So, the value of cosz is -3/5.

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Step-by-step explanation:

Given

5c + 9 - 2c - 7

= 3c + 2

Hope it will help :)❤

The mass is defined as the quantity of matter that a body could have. It is different from the weight, because the weight is a force itself. Usually at the SI base unit, the mass is measured in kilograms, which is denoted by "kg".

Now, the density is defined as the quantity of mass per unit volume. So in math therms it tells you how many mass could you have in a defined unit volume, in other words

where m denotes the mass of the object and v the unit volume. So, for example, you could find that the density of the water is

then it is saying you that if you fill with water a cubic whose sides are each one of 1 meter, then it will have a mass of 997kg.

Remember that given the mass and the volume of an object to get its density, you only need to divide the mass by the unit volume.

Answer:W

Step-by-step explanation:

Got it right

Answer: x= 3500+ x^2

Step-by-step explanation:

x^2 is the formula for an exponential number. Add 3500 because that’s what you started with and then just add x^2

Figure O is reflected followed by a translation of 4 units in the left direction.

Given that:

Figure O and Figure P are shown on the graph.

The translation does not change the shape and size of the geometry. But changes the location.

Figure O is translated leftward by 4 units.

The reflection does not change the shape and size of the geometry. But flipped the image. A reflection is a transformation that maps every point P over a line such that the line segment PP' will intersect the line of reflection at a right angle.

The translated figure O is reflected across the x-axis.

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Answer:

y = 5^(x+1)

Step-by-step explanation:

y = 5^(x+1)

x = 1 for the initial step

Answer:

Step-by-step explanation:

We are given that line a and b and perpendicular to line x. By definition of perpendicular lines, lines a and b form right angles at the intersection of a and x and b and x. By definition, a right angle is equal to 90 degrees, and the sum of angle 2 and 3 is equal 180 degrees, so by the Converse of Same-Side Interior Angle Theorem, line a is parallel to line b.

Note: I don't have a drawing, but we can assume that angles 2 and 3 are same-side interior.

Line m and line n have corresponding angles that are congruent, which implies that line m is parallel to line n.

Theorem: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

Proof:

Let's assume we have a plane with three lines: line l, line m, and line n. We are given that both line m and line n are perpendicular to line l. We want to prove that line m is parallel to line n.

Since line m is perpendicular to line l, the angle formed between line m and line l is 90 degrees. Let's call this angle A.

Similarly, since line n is perpendicular to line l, the angle formed between line n and line l is also 90 degrees. Let's call this angle B.

By the definition of parallel lines, if corresponding angles formed by a transversal and two other lines are congruent, then the two lines are parallel.

In our case, angle A is congruent to angle B (both are 90 degrees). Therefore, line m and line n have corresponding angles that are congruent, which implies that line m is parallel to line n.

Hence, we have proven that in a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

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The volume of the shape is 490cm³

A prism is a solid shape that is bound on all its sides by plane faces. prism is named after the shape of these bases.

The base of this prism is rectangular. The volume of a prism is expressed as;

V = base area × height

base area = 7 × 10 = 70cm²

height = 7cm

the volume of the shape = 70× 7 = 490cm²

therefore the volume of the shape is 490cm³.

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Both planes have a speed of 340 miles per hour and 510 miles per hour, respectively.

In its most basic form, an equation is a mathematical statement that indicates that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign. A mathematical statement made up of two expressions joined by an equal sign is known as an equation. 3x - 5 = 16 is an example of an equation. We get the value of the variable x as x = 7 after solving this equation.

Here,

Let tc be the time of commercial jet and tp be the time of private airplane.

Let vc be the speed of commercial jet and vp be the speed of private airplane.

Let dc be the distance of commercial jet and dp be the distance of private airplane.

now,

dc=dp

vc=vp+170

vc*tc=vp*tp

(vp+170)*1.2=vp*1.8

(vp+170)*2=vp*3

2vp+340=3vp

vp=340 miles/hour

vc=340+170

vc=510 miles/ hour

The speed, in miles per hour, of both airplanes is 340 miles per hour and 510 miles per hour.

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The equation of the line with yellow point in slope intercept form is y = 7 / 6 x + 2.

The equation of a straight line can be represented in different form such as slope intercept form, point slope form, etc.

Therefore, let's represent the line with yellow point in slope intercept form.

Hence,

y = mx + b

where

Therefore, using (0, 2) and (6, 9).

m = 9 - 2 / 6 - 0

m = 7 / 6

Hence, let's find the y-intercept

y = 7 / 6 x + b

2 = 7 / 6(0) + b

b = 2

Therefore,

y = 7 / 6 x + 2.

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9514 1404 393

Answer:

  yes

Step-by-step explanation:

Yes, you can find the distance between two lines using the formula for the distance to a line.

In general, the notion of distance between two lines only make sense when the lines are parallel.

The usual formula for the distance between a point (x, y) and the line ...

  ax +by +c = 0

is ...

  d = |ax +by +c|/√(a^2 +b^2)

Two lines that are parallel will differ only in their values of 'c'. That is, if the two lines are ...

we can arrive at the formula by considering a point (x, y) such that it satisfies the equation for line 2: ax +by +c2 = 0. The distance of that point from line 1 will be ...

  d = |ax +by +c1|/√(a^2 +b^2)

Since we can add or subtract 0 from anything without changing its value, we choose to subtract 0 from the numerator:

  d = |(ax +by +c1) -(ax +by +c2)|/√(a^2 +b^2)

Simplifying gives ...

  d = |c1 -c2|/√(a^2 +b^2)

From the given information:

Batting average of the player = 0.25.

This means that, on average, the player gets 25 hits in 100 times at-bat.

Therefore:

The number of hits which he would expect to get in 360 times at-bat

= Batting Average X Number of Times at-bat

The baseball player would expect to get 90 hits in 360 times at-bat.

Another approach is to use ratio.

Cross multiply

Divide both sides by 100 to solve for x

Therefore, the baseball player would expect to get 90 hits in 360 times at-bat.

<GEC=<HFA

As

So

Cancel FE

Also

<CAH=<GCE(Alternative interior angles)

Henceforth

△GEC ≅ △HFA(ASA)

Answer:

2/3

Step-by-step explanation:

8+6=14

14/21=2/3

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When writing a proof , by copying the “given” statement(s) from the original problem

Given data ,

A true assertion that is provided in the problem or is known to be true should be the first statement in a proof. The subsequent logical argument has this as its foundation.

We can provide the groundwork for the proof and proceed to the desired conclusion by duplicating the "given" statement(s) from the original problem.

Once the first statement is established, we can move on to writing additional statements that are each logically supported by the first statement.

The logical evolution of the preceding assertions demonstrates that the last statement should be the intended conclusion.

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Answer:

R is an equivalence relation, since R is reflexive, symmetric, and transitive.

Step-by-step explanation:

The relation R is an equivalence if it is reflexive, symmetric and transitive.

The order to options required to show that R is an equivalence relation are;

((a, b), (a, b)) ∈ R since a·b = b·a

Therefore, R is reflexive

If ((a, b), (c, d)) ∈ R then a·d = b·c, which gives c·b = d·a, then ((c, d), (a, b)) ∈ R

Therefore, R is symmetric

If ((c, d), (e, f)) ∈ R, and ((a, b), (c, d)) ∈ R therefore, c·f = d·e, and a·d = b·c

Multiplying gives, a·f·c·d = b·e·c·d, which gives, a·f = b·e, then ((a, b), (e, f)) ∈R

Therefore R is transitive

From the above proofs, the relation R is reflexive, symmetric, and transitive, therefore, R is an equivalent relation.

Reasons:

Prove that the relation R is reflexive

Reflexive property is a property is the property that a number has a value that it posses (it is equal to itself)

The given relation is ((a, b), (c, d)) ∈ R if and only if a·d = b·c

By multiplication property of equality; a·b = b·a

Therefore;

((a, b), (a, b)) ∈ R

The relation, R, is reflexive.

Prove that the relation, R, is symmetric

Given that if ((a, b), (c, d)) ∈ R then we have, a·d = b·c

Therefore, c·b = d·a implies ((c, d), (a, b)) ∈ R

((a, b), (c, d)) and ((c, d), (a, b)) are symmetric.

Therefore, the relation, R, is symmetric.

Prove that R is transitive

Symbolically, transitive property is as follows; If x = y, and y = z, then x = z

From the given relation, ((a, b), (c, d)) ∈ R, then a·d = b·c

Therefore, ((c, d), (e, f)) ∈ R, then c·f = d·e

By multiplication, a·d × c·f = b·c × d·e

a·d·c·f = b·c·d·e

Therefore;

a·f·c·d = b·e·c·d

a·f = b·e

Which gives;

((a, b), (e, f)) ∈ R, therefore, the relation, R, is transitive.

Therefore;

R is an equivalence relation, since R is reflexive, symmetric, and transitive.

Based on a similar question posted online, it is required to rank the given options in the order to show that R is an equivalence relation.

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