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Find the area of triangle ABC with vertices A(2, 3), B(1, -3), and C(-3, 1).
a.
10 units2
c.
14 units2
b.
12 units2
d.
16 units2

Answer:

  below the line

Step-by-step explanation:

The "shading" is the blue/purple color. The graph is shaded below the line y=-2x+6, as it should be when y is less than or equal to -2x + 6.

(Values of y get smaller in the downward direction, to for any given value, ones that are less than that will be below it.)

2x - 3 = -8

    +3     + 3

2x = -5

/2    /2

x = -2.5 (Answer)

Prove.

2(-2.5) - 3 = -8

-5 - 3 = -8

-8 = -8

True.

Answer:

Answer -2.5

Step-by-step explanation:

Simple:

2x-3=-8

   +3  +3,  Add 3  on both sides of the equal sign.

2x = -5, Bring the 2x down and add the -8 + 3

2        2,  Divide by 2

x= -2.5

Hope my answer has helped you!

Answer: Option D

Step-by-step explanation:

You know that the ratio between the radii of the two sphere is 3:5. Knowing this fact, you can calculate the ratio of their volume with this procedure:

[tex]ratio\ volume=\frac{3^3}{5^3}[/tex]

You need to remember that:

[tex]a^3=a*a*a[/tex]

 Then, you can rewrite it as:

[tex]ratio\ volume=\frac{3*3*3}{5*5*5}[/tex]

Finally, you get the the ratio of the volumes of the spheres is:

[tex]ratio\ volume=\frac{27}{125}[/tex] or 27:125

This matches with the option D.

Answer:

The ratio of their volumes is 27 : 125

Step-by-step explanation:

Points to remember

Volume of sphere V = 4/3(πr³)

Where r is the radius of sphere

To find the ratio of volume of spheres

It is given that,  the ratio between the radii of the two sphere is 3:5

V₁ =  4/3(πr₁³)  =  4/3(π3³) and

V₂ =  4/3(πr₂³) =  4/3(π5³)

V₁/V₂ = 4/3(π3³)/4/3(π5³)

 = 3³/5³ = 27/125

Therefore  the ratio of their volumes = 27 : 125

Answer:

$108

Step-by-step explanation:

Step 1: Find the interest. You do this by multiplying the amount borrowed by the interest rate. In this case, you multiply 1,200 by 9% (0.09). 1,200 * 0.09 is $108. That's the amount she paid on the loan in her first year.

Answer:

36π cubic inches

Step-by-step explanation:

The formula for the volume of a sphere is ...

V = (4/3)πr^3

Fill in the given value and simplify:

V = (4/3)π(3 in)^3 = 36π in^3

First we pick the equation. Let's say we pick first one. From it we express y.

[tex]-5x+y=-2\Longrightarrow y=-2+5x[/tex]

Then we use this y and plug it in instead of y in the second equation.

[tex]-3x+6(-2+5x)=-12[/tex]

Now just solve for x.

[tex]

-3x-12+30x=-12 \\

-3x=-30x \\

\underline{x=10}

[/tex]

We plug this x in the first equation.

[tex]-5\cdot10+y=-2[/tex]

And solve for y.

[tex]

-50+y=-2 \\

\underline{y=48}

[/tex]

So the solution to the equation is geometrically a point P which lies on the intersection of the two lines.

[tex]\boxed{P(10,48)}[/tex]

Hope this helps.

r3t40

Answer:

52

Step-by-step explanation:

add add add add add add adddaddd addda

The inscribed angle is half that of the arc it comprises. The measure of both the angle is 52°.

we know that the inscribed angle is half that of the arc it comprises.

Here, the arc that the inscribed angles comprise is the same.

2(3m+2)° = (4m+20)°

Solve for m

6m + 4 = 4m + 20

6m - 4m = 20 - 4

2m = 16

m = 8

To find the measure of the angle

(4m+20)°= 4(8) + 20 = 52°

2(3m+2)° = 2(26) = 52

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This is a problem of projectile motion. A projectile is an object you throw with an initial velocity and whose trajectory is determined by the effect of gravitational acceleration. The general equation in this case is described as:

[tex]h(t)=-\frac{1}{2}gt^2+v_{0}t+h_{0}[/tex]

Where:

[tex]h(t): \ height \ at \ any \ time \\ \\ g: \ acceleration \ due \ to \ gravity \ 9.8m/s^2 \ or \ 32.16ft/s^2 \\ \\ v_{0}= \ Initial \ velocity[/tex]

So:

[tex]v_{0}=64ft/s \\ \\ h_{0}=3ft[/tex]

Finally, the equation is:

[tex]h(t)=-\frac{1}{2}(32.16)t^2+(64)t+3 \\ \\ \boxed{h(t)=-16.08t^2+64t+3}[/tex]

The rocket will reach the maximum height at the vertex of the parabola described by the equation [tex]h(t)=-16.08t^2+64t+3[/tex]. Therefore, our goal is to find [tex]t[/tex] at this point. In math, a parabola is described by the quadratic function:

[tex]f(x)=ax^2+bx+c[/tex]

So the x-coordinate of the vertex can be calculated as:

[tex]x=-\frac{b}{2a}[/tex]

From our equation:

[tex]a=-16.08 \\ \\ b=64 \\ \\ c=3[/tex]

So:

[tex]t=-\frac{64}{2(-16.08)} \\ \\ \boxed{t=1.99s}[/tex]

So the rocket will take its maximum value after 1.99 seconds.

From the previous solution, we know that after 1.99 seconds, the rocket will reach its maximum, so it is obvious that the maximum height is given by [tex]h(1.99)[/tex]. Thus, we can find this as follows:

[tex]H_{max}=h(1.99)=-16.08(1.99)^2+64(1.99)+3 \\ \\ \boxed{H_{max}=66.68ft}[/tex]

So the maximum height the rocket will reach is 66.68ft

The rocket is in the air until it hits the ground. This can be found setting [tex]h(t)=0[/tex], so:

[tex]0=-16.08t^2+64t+3 \\ \\ Applying \ quadratic \ formula: \\ \\ t_{12}=\frac{-b \pm \sqrt{b^2-4ac}}{2a} \\ \\ a=-16.08 \\ \\ b=64 \\ \\ c=3 \\ \\ t_{12}=\frac{-64 \pm \sqrt{64^2-4(-16.08)(3)}}{2(-16.08)} \\ \\ t_{1}=4.0264 \\ \\ t_{2}=-0.046[/tex]

We can't have negative value of time, so the only correct option is [tex]t_{1}=4.0264[/tex] and rounding to the nearest hundredth we have definitively:

[tex]\boxed{t=4.03s}[/tex]

The expression x + 7 will determine how many cats Staci has.

Since Staci has 7 more cats than Taylor (x), we know the expression uses addition.

25-17=8

8÷2=4

4 apples per friend.

Answer:

the slope is 1/2

Work:

if the x-coordinate is two times the y-coordinate, that means you are going up one over two each time

slope= y2-y1/x2-x1

so you can pick two points, like (2, 1) and (4,2) and use the formula

slope= 2-1/4-2= 1/2

Answer:

(7, - 4) and r = 7

Step-by-step explanation:

The equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k) are the coordinates of the centre and r is the radius

(x - 7)² + (y + 4)² = 49 ← is in standard form

with (h, k) = (7, - 4 ) and r = [tex]\sqrt{49}[/tex] = 7

ANSWER

B. 24 sq. units.

EXPLANATION

The area of a triangle is calculated using the formula:

[tex]Area = \frac{1}{2} \times AB \times BC[/tex]

From the graph ,

AB=3--3=6 units.

BC=3--5=8 units

We substitute the values into the formula to get,

[tex]Area = \frac{1}{2} \times 6 \times 8 = 24sq. \: units[/tex]

The correct answer is B

Answer:

  see below

Step-by-step explanation:

The equation is in point-slope form, so you know the line goes through the point (x, y) = (-3, 2) and it has a slope of -3. The slope tells you the line goes down 3 units for each 2 units to the right.

Answer:

not safe

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds you of the relationship between an angle and the ratio of side opposite to hypotenuse:

Sin = Opposite/Hypotenuse

You have the ratio of lengths and you want to find the angle. You use the inverse sine function (arcsin, or sin⁻¹) for this purpose.

sin(angle with ground) = (16.5 ft)/(17 ft) = 33/34

angle with ground = arcsin(33/34) ≈ 76.1°

This angle is greater than the angle of 70° recommended as a maximum. The ladder will not be safe at this height.

_____

A calculator does all the work. You just need to make sure it is in degrees mode.

Answer:

7√2 / 2

Step-by-step explanation:

An isosceles right triangle is also known as a 45-45-90 triangle.  For such a triangle, the hypotenuse (long side) is √2 the legs.

s√2 = 7

s = 7/√2

s = 7√2 / 2

You can also show this with Pythagorean theorem.

The graph of f(x) = [tex]2^{x+3}[/tex] is option "B" .

An exponential function is a mathematical function of the following form:

f ( x ) = [tex]a^{x}[/tex]. where x is a variable, and a is a constant called the base of the function. The most commonly encountered exponential-function base is the transcendental number e , which is equal to approximately 2.71828.

According to the question

Represents the graph of f(x) = [tex]2^{x+3}[/tex]

f(x) = [tex]2^{x+3}[/tex] is a  exponential function

To draw the graph of f(x) = [tex]2^{x+3}[/tex]

The values will be

x                  f(x)

0                   8

1                   16

-1                   4

-2                  2

-3                  0

Hence, the graph of f(x) = [tex]2^{x+3}[/tex] is option "B" .

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The graph of the function represented by the equation f(x) = 2ˣ + 3 is (c)

From the question, we have the following parameters that can be used in our computation:

The function

Where, we have

f(x) = 2ˣ + 3

The above function is an exponential function that has been transformed as follows

Using the above as a guide, we have the following:

The graph of the function is (c)

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A relative minimum = −36 , relative maximum = 64

Answer:

A

Step-by-step explanation:

The relative minimum is where the curve changes directions from down to up.  The relative maximum is where the curve changes direction from up to down.

In the interval [-4, 4], there are two relative minimums: y=-36.  And there is one relative maximum: y=64.

Answer A.

Answer:

54°

Step-by-step explanation:

y is an inscribed angle.  That means that it is half the measure of the arc it intercepts.  The arc it intercepts is 108, so the angle is 54°

Check the picture below.

Answer:

g(x) has an axis of symmetry at x = 3.

g(x) is shifted right 3 units from the graph of f(x)

g(x) is shifted up 4 units from the graph of f(x).

Step-by-step explanation:

The parent function is:

[tex]f(x)=x^2[/tex]

The transformed function is [tex]g(x)=-x^2+6x-5[/tex].

This new function can be rewritten in the vertex form as:

[tex]g(x)=-(x-3)^2+4[/tex]

This function is obtained by shifting the parent function 3 units right and 4 units up.

The axis of symmetry is x=3;(x=h) and h=3.

There is no horizontal stretch or compression.

The new function is however reflected in the x-axis

Step-by-step explanation:

on edge its A,C,D :)

For this case we have the following system of equations:

[tex]4x + 5y = -4\\4x + 3y = 12[/tex]

We multiply the second equation by -1:

[tex]-4x-3y = -12[/tex]

We add the equations:

[tex]4x-4x + 5y-3y = -4-12\\5y-3y = -4-12\\2y = -16\\y = \frac {-16} {2}\\y = -8[/tex]

We find the value of "x":

[tex]4x = 12-3y\\x = \frac {12-3y} {4}\\x = \frac {12-3 (-8)} {4}\\x = \frac {12 + 24} {4}\\x = \frac {36} {4}\\x = 9[/tex]

Thus, the solution of the system is given by:

[tex](x, y) :( 9, -8)[/tex]

ANswer:

(9, -8)

Answer:

x = 9 and y = -8

Step-by-step explanation:

It is given that,

4x + 5y = -4     ----(1)

4x + 3y = 12    -----(2)

To find the value of x and y

eq (1) - eq (2) ⇒

4x + 5y = -4     ----(1)

4x + 3y = 12    -----(2)

  0 + 2y = -16

y = -16/2 = -8

Substitute the value of y in eq (1)

4x + 5y = -4     ----(1)

4x + 5*-8 = -4

4x - 40 = -4

4x = -4 + 40 = 36

x = 36/4 = 9

x = 9 and y = -8

Answer:

  97π m² ≈ 304.7 m²

Step-by-step explanation:

The area of a circle is given by the formula ...

  A = πr²

The sum of the areas of the two circles is ...

  A = π(9 m)² + π(4 m)² = π(81 +16) m²

  = 97π m² ≈ 304.7 m²

Answer:

f(x) = (x - 1)(x + 2)(x - 3)

Step-by-step explanation:

We are given the following function and we are to factorize it completely:

[tex] f ( x ) = x ^ 3 - 2 x ^ 2 - 5 x + 6 [/tex]

To factorize this completely, we will use the rational roots theorem.

[tex] x ^ 3 - 2 x ^ 2 - 5 x + 6 [/tex]

P = ± multiples of constant term [tex]6 = \pm1, \pm2, \pm3, \pm6[/tex]

Q = ± multiples of the coefficient of highest degree term  [tex] = \pm1[/tex]

So the factors will be [tex]\frac{P}{Q}[/tex].

The possible rational roots are [tex] 1, \pm2, \pm3, \pm6[/tex].

1 is a confirmed root and now we will use synthetic division to find the other rational roots:

1 | 1  -2  -5   6

        1   -1   -6

___________

   1   -1   -6   0    

So the polynomial will be [tex](x^2 - x - 6)[/tex] which can we factorize now.

[tex]x^2 - x - 6 = x^2 - 3x + 2x - 6[/tex]

[tex]x(x - 3) + 2(x - 3) = (x+2)(x-3)[/tex]

Therefore, the completely factorized form of the given function is f(x) = (x - 1)(x + 2)(x - 3).

Answer:

A)SAS

Step-by-step explanation:

The tick marks represent which sides of the triangles are congruent, while the arc represents which angles are congruent.

Since the arc is in between the sides that are congruent therefore the postulate is Side Angle Side

Without visual or informational context, the congruence criteria between the pairs of triangles cannot be determined. It's essential to know specific details about the triangles to apply congruence criteria like Side-Angle-Side (SAS), Angle-Side-Angle (ASA), or Angle-Angle-Side (AAS). However, with no further details, the answer is 'D. None'.

To determine which criterion for triangle congruence can be used to prove that the pair of triangles are equal, we first need to examine the triangles. SAS refers to Side-Angle-Side, which is when two sides and the included angle of one triangle is congruent to two sides and the included angle of another triangle. ASA refers to Angle-Side-Angle, which is when two angles and the included side of one triangle is congruent to two angles and the included side of another triangle. AAS refers to Angle-Angle-Side, which is when two angles and a non-included side of one triangle is congruent to two angles and a non-included side of another triangle. So, without a visual or further information about the triangles, we cannot directly apply any particular congruence criteria and the answer would be 'D. none'. However, with more information, the pair of triangles can possibly be proven congruent using either SAS, ASA, or AAS methods.

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Answer:

Step-by-step explanation:

In the equation, 72t represents the initial upwards velocity and 5 represents the initial launching height.  The leading term represents the pull of gravity on the object in the English system of measurement.

So the first question says the initial height is 5 feet.  TRUE

The second question says the initial vertical velocity is -72.  FALSE (it's positive 72 ft/sec)

The third question says that the object will hit the ground after approximately 4.57 seconds.  TRUE.  Find this by setting the h(t) on the left equal to 0, since this is the height at any time during the flight.  When h(t) = 0, that means that there is NO height, which means the object is on the ground.  Set the equation equal to 0 and factor to find t.  Putting that into the quadratic formula gives you t values of -.068 and 4.57.  Since the 2 things in math that will NEVER EVER be negative are distances and time, we can safely disregard the negative t value and go with t = 4.57.

The fourth question says that after t = 3 seconds, the object is 173 feet high.  FALSE.  Find this by subbing in a 3 for every t in the equation.

[tex]h(3)=-16(3)^2+72(3)+5[/tex]

This gives you that h(3) = 77 feet

The last question says that at t = 0, h(t) = 0.  FALSE again.  Sub in a 0 for every t in the equation, and you get h(0) = 5, which is the initial launching height.

Answer:

The initial height is 5 feet.

The object will hit the ground after approximately 4.57 seconds.

A & C

Step-by-step explanation

An object is launched into the air. The projectile motion of the object can be modeled using the function h(t) = -16t^2 + 72t + 5, where t is the time in seconds since the launch and h(t) represents the height in feet of the object after t seconds

General equation is

V_0 is the initial velocity

h_0 is the initial height

From the given equation , the initial height is 5 feet

Initial velocity is +72 feet / sec

When the onbject hits the ground, the height becomes 0

So we plug in 0 for h(t) and solve for t

USe quadratic formula to solve for t

a=-16, b= 72, c= 5

t= -0.06  and t= 4.568

The object will hit the ground after approximately 4.57 seconds.

To find out the height after 3 seconds, plug in 3 for t

At t=0 , h(0) = 5

Answer:

The Answer is multiple choice and the answer is B. Only B.

Step-by-step explanation:

The Answer is multiple choice and the answer is B.

We have given that,

St. Louis, Missouri, and Washington, D.C.

Have very similar latitudes. (Latitude measures how far north or south a place is, which in turn affects the temperature.)

The latitude is specified by degrees, starting from 0° and ending up with 90° to both sides of the equator, making latitudes Northern and Southern.

The box plots below show the daily high temperatures (in degrees Celsius) that were recorded during July 201320132013 in each city.

The pieces of the information that can be gathered from these box plots are option is B.

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Answer:

Step-by-step explanation:

All other things being equal, spinner A is fine. It is a fair spinner.

Spinner B is not fair. Players 1 and 3 have only 1 number each. Player 2 on the other hand, has 2 numbers that work for him. If player 2 puts in two dollars and players 1 and 3 one dollar each, that should even up the odds. Now you want it just to be fair. So I think player 2 has to put up 2 dollars and players 1 and 3 each put up one. The pot is 4 dollars each time it is played.

Spinner 3 is not fair either. Player one has 2 chances. Player 2 has 3 chances and player 3 has but one chance. There are 6 chances in all. Player 1 should put up 2 dollars to play player 1 should put up 3 dollars and player 1 should put up 1 dollar.

The travel of the barnacle would be the circumference of the circle.

Circumference formula is 2 x PI x r

Circumference  = 2 x PI x 1m

= 2 x PI

If you use 3.14 for PI the circumference is 2 x 3.14 = 6.28 m.

Round the answer as needed.

I think that the first choice is the correct one

Answer:

a. an = -1 • 4^(n - 1)

Step-by-step explanation:

It's easy see that each term is multiplied by 4 to get the next term, so the common ratio will be 4, that reduces the choices to answers A and B (since C and D use -1 as common ratio).  Also, normally a geometric progression uses "n-1", so we'll first try with A to see if it works.

a1 = (-1) * 4^0 = -1 * 1 = -1

a2 = (-1) * 4^1 = -1 * 4 = -4

a3 = (-1) * 4^2 = -1 * 16 = -16

Yes, it works... so the answer is an = -1 • 4^(n - 1)