- 1 (multiplicity 3), 3 (multiplicity 4)

y = (x +2)(x -1)(x -3) . . . . or . . . . y = x³ -2x² -5x +6

The graph shows y=0 at x=-2, x=1, and x=3. These are called the "zeros" or "roots" of the function, because the value of the function is zero there.

When "a" is a zero of a polynomial function, (x -a) is a factor. This means the factors of the graphed function are (x -(-2)), (x -1) and (x -3). The function can be written as the product of these factors:

... y = (x +2)(x -1)(x -3) . . . . . the equation represented by the graph

Or, the product can be multiplied out

... y = (x +2)(x² -4x +3)

... y = x³ -2x² -5x +6 . . . . . the equation represented by the graph

By using two equations, 'Thomas's age = Huilan's age + 15' and 'Thomas's age + Huilan's age = 33', and substituting Huilan's age, we find that Thomas is 24 years old.

To solve this problem, we use a system of linear equations. Based on the given information, we can create two equations:

Replacing Huilan's age in the second equation with '(Thomas's age - 15)', we get: Thomas's age + (Thomas's age - 15) = 33

This simplifies to 2 * Thomas's age = 48, meaning Thomas's age = 24 years.

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f(x) = 3·2^x

When x=0, any exponential term will have a value of 1, so the y-intercept is the multiplier of the exponential function. Here, it is 3.

When x=1, the exponential term will have a value equal to its base, so the multiplier just found will be multiplied by the base value. Here, f(1) = 3·2, so the base of the exponential term is 2.

Given these considerations, the function is ...

... f(x) = 3·2^x

_____

Comment on notation

The caret (^) is used to signify an exponent. When the exponent consists of anything other than a single number or variable, it must be put in parentheses: 2^(1/2), for example.

The expressions you have written all look like linear functions.

Answer:

C. linear

Step-by-step explanation:

Because they spend $8 to manufacture a basketball every time and stays constant, the rate of cost growth at the factor is linear.

If the factory manufactures basketball spends $8 on each basketball it produces, then  the rate of cost growth at the factory will be  neither linear nor nonlinear.

Let 'x' be the total number of basket ball produced by the factory.

The amount paid for the each ball is $8.

Let 'y' be the total amount spent on the manufacturing of the basketball.

Then,

y = 8x.

As, the total manufacturing cost of the basketball depends upon the total production of the basketball.

Thus, total amount 'y' is the function of 'x'.

Differentiating 'y' with respect to 'x',

[tex]\frac{dy}{dx} =\frac{d(3x)}{dx}[/tex]

[tex]\frac{dy}{dx} =3\frac{dx}{dx} =3[/tex]

As, 3 is the constant value, it means change in the cost with respect to manufacturing of basketball is constant.

Therefore, the rate of cost growth at the factory is neither linear nor nonlinear.

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45° and 135°

If the smaller is represented by x, then the larger is 3x. The two angles are supplementary, so ...

... x + 3x = 180°

... 4x = 180°

... 180°/4 = x = 45°

... 3x = 135°

The two angles are 45° and 135°.

Answer:

45° and 135°

Step-by-step explanation:

If the smaller is represented by x, then the larger is 3x. The two angles are supplementary, so ...

... x + 3x = 180°

... 4x = 180°

... 180°/4 = x = 45°

... 3x = 135°

The two angles are 45° and 135°.

Step-by-step explanation:

Answer:

15

Step-by-step explanation:

Answer:

216 in squared

Step-by-step explanation:

using the formula, a cube has six sides so 6 squared is 36 times 6 equals to 216

Answer:

65550 is the population at the end of 2017

Step-by-step explanation:

Population at the beginning of 2106:  60000

Increased 15% in 2016

Increase = 60000*.15 = 9000

New population = 60000+9000 = 69000

The population at the beginning of 2017: 69000

Decrease by 5% in 2017

Decrease 69000*.05 = 69000*.05 = 3450

New population = 69000-3450=65550

The population at the end of 2017 is 65550

Answer:

[tex]a=6[/tex], [tex]b=6\sqrt{2}[/tex], [tex]c=2\sqrt{3}[/tex], and [tex]d=6[/tex]

Step-by-step explanation:

Looking at the left triangle, we can solve for [tex]a[/tex] and [tex]c[/tex].

a:

[tex]a[/tex] is to the opposite side of 60° angle, also we know the hypotenuse, [tex]4\sqrt{3}[/tex]. The ratio that relates opposite with hypotenuse is SINE. Thus we can write:

[tex]sin(A)=\frac{opposite}{hypotenuse}\\sin(60)=\frac{a}{4\sqrt{3}}\\[/tex]

Cross multiplying and solving for [tex]a[/tex]:

[tex]sin(60)=\frac{a}{4\sqrt{3} }\\a=sin(60)*4\sqrt{3}\\a=\frac{\sqrt{3}}{2}*4\sqrt{3}\\a=\frac{12}{2}=6[/tex]

( we know [tex]sin(60)=\frac{\sqrt{3}}{2}[/tex] and also [tex]\sqrt{a}*\sqrt{a}=a[/tex] )

c:

[tex]c[/tex] is to the adjacent side of 60° angle, also we know the hypotenuse, [tex]4\sqrt{3}[/tex]. The ratio that relates adjacent with hypotenuse is COS. Thus we can write:

[tex]cos(A)=\frac{adjacent}{hypotenuse}\\cos(60)=\frac{c}{4\sqrt{3}}[/tex]

Cross multiplying and solving for [tex]c[/tex]:

[tex]cos(60)=\frac{c}{4\sqrt{3}}\\c=cos(60)*4\sqrt{3}\\c=\frac{1}{2}*4\sqrt{3}\\c=2\sqrt{3}[/tex]

( we know [tex]cos(60)=\frac{1}{2}[/tex] )

Looking at the triangle to the right, we can solve for [tex]b[/tex] and [tex]d[/tex].

b:

[tex]a[/tex] is to the opposite side of 45° angle. We have figured out that [tex]a=6[/tex]. Also we know that [tex]b[/tex] is the hypotenuse.The ratio that relates opposite with hypotenuse is SINE. Thus we can write:

[tex]sin(A)=\frac{opposite}{hypotenuse}\\sin(45)=\frac{6}{b}\\b=\frac{6}{sin(45)}\\b=\frac{6}{\frac{1}{\sqrt{2}}}\\b=6*\frac{\sqrt{2}}{1}\\b=6\sqrt{2}[/tex]

( we know [tex]sin(45)=\frac{1}{\sqrt{2}}[/tex] )

d:

To solve for [tex]d[/tex], we can use the pythagorean theorem. Given by:

[tex]a^2+b^2=c^2[/tex]

Where,

In the triangle on the right, [tex]b[/tex] is the hypotenuse and [tex]a[/tex] and [tex]d[/tex] are the two legs. Using pythagorean theorem and solving for [tex]d[/tex], we get:

[tex]d^2+a^2=b^2\\d^2+(6)^2=(6\sqrt{2})^2\\d^2+36=72\\d^2=72-36\\d^2=36\\d=6[/tex]

( we know that [tex]\sqrt{a}*\sqrt{a}=a[/tex] )

Looking at the answers, 2nd answer choice is right.

The function you're to evaluate is

... f(x) = 0.5x +3.40

Using function notation, the value for x = 2 is ...

... f(2)

For x = 3.5, the value is ...

... f(3.5)

and so on.

To actually evaluate the function, you need to put the value where x is in the function definition and do the arithmetic:

... f(2) = 0.5·2 + 3.40 = 1.00 +3.40 = 4.40

_____

The function gives the charge for a package based on its weight in ounces. The result of evaluating the function for the given weights is that you find the charges for delivery of those packages.

Answer:

He can have 6 friends eat cake

Step-by-step explanation:

If each friend can eat 1/6 of the cake, we need to find how many friend can have a piece

1/6 * f = 1 cake

Multiply each side by 6

6*1/6 * f = 1 *6

f = 6

He can have 6 friends eat cake

Answer:

6 friends can have a piece of cake.

Step-by-step explanation:

Each friend eats 1/6 of the cake.

There are six 1/6's in a unit.

6 friends can have a piece of cake.

Answer:

Yes

Step-by-step explanation:

3+5 is 8 and 6+2 is also 8. You can draw 3 apples with 5 other apples, and count that there are 8 of them and draw 6 apples with 2 other apples that make up 8 apples. Or you can rearrange the apples to be 5 and 3 and 6 and 2.

Answer:

False

Step-by-step explanation:

It is the exponent of a base.

The variable in an exponential function is always the exponent of the power is false.

When the expression of function is such that it involves the input to be present as an exponent (power) of some constant, then such function is called exponential function.

Their usual form is specified below. They are written in several such equivalent forms.

The variable in an exponential function is always the exponent of the power is false. It is the exponent of a base.

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Answer:

41 words per minute

Step-by-step explanation:

You just need to divide words by minutes and that will give you words per minute.

410 words / 10 minutes = 410 / 10 words per minute = 41 words per minute

Answer:

A direct variation equation to represent this ;  [tex]y = \frac{2}{3}x[/tex]

Step-by-step explanation:

Direct Variation states that a relationship between two variables in which one is a constant multiple of the other one.

In other words, when one of the variable changes then the other changes in proportion to the first.

If b is directly proportional to a, then the equation is in the i.e,

form [tex]b = ka[/tex] where k is the constant of variation.

Let y represents the number of muffins and x represents the amount of flour.

It is given that the number of muffins varies directly with the amount of flour you use.

As per the given statement:

y = 2 dozen corn and x = 3 cups of flour.

Then, by definition of Direct variation;

y = kx

Substitute the given values to find k;

[tex]2 = 3k[/tex]

Divide both sides by 3 we get;

[tex]k = \frac{2}{3}[/tex]

then, equation is, [tex]y = \frac{2}{3}x[/tex]

Therefore, a direct variation equation to represent this situation is; [tex]y = \frac{2}{3}x[/tex]

Answer:

17.9

Step-by-step explanation:

Here we are given a right angled triangle with a known angle of 72°, length of the perpendicular to be 17 feet and we are to find the length of the hypotenuse x.

For that, we can use the formula for sin for which we need an angle and the lengths of base and hypotenuse.

[tex]sin \alpha =\frac{perpendicular}{x}[/tex]

So putting in the given values to get:

[tex]sin 72=\frac{17}{x} [/tex]

[tex]x=\frac{17}{sin 72}[/tex]

[tex]x=17.8[/tex]

Therefore, the length of six cars is the closest to 17.9.

Answer:

c)No, P is the distance from A to B, and Q is the distance from B to A.

Step-by-step explanation:

Point P partitions the directed segment from A to B into a 1:3 ratio.

Ratio is 1:3

So AP is 1  and PB is 3

Q partitions the directed segment from B to A into a 1:3 ratio.

Ratio is 1:3

So BQ is 1  and QA is 3.

That is AQ= 3  and BQ= 1

The ratio of Q and P varies

AP =1  and AQ=3

So P  and Q  are not at the same point.

Because P is the distance from A  to B  and Q is the distance from B to A

Answer:

c

Step-by-step explanation:

bc

Answer:

If c>0, then the solution is imaginary, which is not a real solution.

Step-by-step explanation:

ax^2 + c = 0, with a > 0

Subtract c from each side

ax^2 +c-c = 0-c

ax^2 = -c

Divide each side by a

ax^2/a = -c/a

x^2 = -c/a

Take the square root of each side

sqrt(x^2) = sqrt(-c/a)

AAAAAAAH

The only way to have real square roots is for -c/a to be positive.

We know that a>0, so -c >0

-c>0

Divide each side by -1, remembering to flip the inequality

c<0

If c<0  we have real solutions

If c=0  then x=0 which is a real solution

If c>0, then the solution is imaginary, which is not a real solution.

Answer:

D. differences

Step-by-step explanation:

One way to identify a linear function is by checking the differences over equal intervals. If they are the same, then the function is linear.

Answer:

D. Differences

Step-by-step explanation:

Linear function grow by equal differences over equal intervals as, on a graph, it is shown by a steady increasing line. Exponential functions, on the other hand, do not.

To find the solution to the system of linear equations, set the equations equal to each other and solve for x. Then substitute this value back into either of the equations to find y.

To find the solution to the system of linear equations, we can set the equations equal to each other:

58x + 18 = -12x - 1

Combining like terms, we get:

70x = -19

Divide both sides by 70:

x = -19/70

Substitute this value back into either of the equations to find y:

y = 58(-19/70) + 18 = -10.2

Therefore, the solution to the system of linear equations is x = -19/70 and y = -10.2.

Answer:

D . 58 miles

Step-by-step explanation:

In order to find the distance travelled (d) in each stage, we will use the following expression.

d = v × t

where,

v: speed

t: time

First stage

v = 70 mi/h

t = 45 min × (1 h / 60 min) = 0.75 h

d = v × t = 70 mi/h × 0.75 h = 53 mi

Second stage

v = 20 mi/h

t = 15 min × (1 h / 60 min) = 0.25 h

d = v × t = 20 mi/h × 0.25 h = 5.0 mi

The total distance tyraveled is 53 mi + 5.0 mi = 58 mi

Josie drove 52.5 miles for the first part of her trip and 5 miles for the second part, totaling 57.5 miles. After rounding, it is approximately 58 miles, so the answer is D.

To calculate the total distance Josie drove, we need to find the distance for each segment of her journey and then sum them up. For the first part of her trip, she drove for 45 minutes at a speed of 70 miles per hour. Since time needs to be in hours to use the formula distance = speed × time, we convert 45 minutes to hours by dividing by 60, giving us 0.75 hours. So, the distance for the first part is 70 miles/hour × 0.75 hours, which equals 52.5 miles.

For the second part of the trip, Josie drove for 15 minutes at 20 miles per hour. Converting 15 minutes to hours, we get 0.25 hours. Therefore, the distance for the second part is 20 miles/hour × 0.25 hours, giving us 5 miles. To find the total distance, we add the distances from both parts of the trip, resulting in 52.5 miles + 5 miles = 57.5 miles.

After rounding to the nearest mile, the total distance Josie drove was roughly 58 miles.

Therefore, the correct answer is D.

Answer:

Congruent; Not Congruent; Congruent; Not Congruent; Not Congruent; Congruent

Step-by-step explanation:

We will use the distance formula to find the length of each segment:

[tex]AB=\sqrt{(-4--3)^2+(4-2)^2}=\sqrt{(-1)^2+(2)^2}=\sqrt{1+4}=\sqrt{5}\\\\BC=\sqrt{(-3--1)^2+(4-1)^2}=\sqrt{(-2)^2+(3)^2}=\sqrt{4+9}=\sqrt{13}\\\\AC=\sqrt{(2-1)^2+(-4--1)^2}=\sqrt{(1)^2+(-3)^2}=\sqrt{1+9}=\sqrt{10}[/tex]

In order for ABC to be congruent to DEF, AB must be congruent to DE, BC must be congruent to EF, and AC must be congruent to DF:

[tex]DE=\sqrt{(4-3)^2+(-2--4)^2}=\sqrt{(1)^2+(2)^2}=\sqrt{1+4}=\sqrt{5}\\\\EF=\sqrt{(3-1)^2+(-4--1)^2}=\sqrt{(2)^2+(-3)^2}=\sqrt{4+9}=\sqrt{13}\\\\DF=\sqrt{(4-1)^2+(-2--1)^2}=\sqrt{(3)^2+(-1)^2}=\sqrt{9+1}=\sqrt{10}[/tex]

Since AB is congruent to DE, BC is congruent to EF and AC is congruent to DF, the two triangles are congruent.

In order for ABC to be congruent to JKL, AB must be congruent to JK, BC must be congruent to KL, and AC must be congruent to JL.  We know the measurements of AB, BC and AC;

[tex]JK=\sqrt{(-4--2)^2+(-1--3)^2}=\sqrt{(-2)^2+(2)^2}=\sqrt{4+4}=\sqrt{8}\\\\KL=\sqrt{(-2--1)^2+(-3-0)^2}=\sqrt{(-1)^2+(-3)^2}=\sqrt{1+9}=\sqrt{10}\\\\JL=\sqrt{(-1--4)^2+(0--1)^2}=\sqrt{(3)^2+(1)^2}=\sqrt{9+1}=\sqrt{10}[/tex]

While AC is congruent to JL, the other two corresponding pairs of sides are not congruent.  Therefore the triangles are not congruent.

In order for ABC to be congruent to QRS, AB must be congruent to QR, BC must be congruent to RS, and AC must be congruent to QS.  We know the measurements of AB, BC and AC;

[tex]QR=\sqrt{(3-4)^2+(3-1)^2}=\sqrt{(-1)^2+(2)^2}=\sqrt{1+4}=\sqrt{5}\\\\RS=\sqrt{(3-1)^2+(3-0)^2}=\sqrt{(2)^2+(3)^2}=\sqrt{4+9}=\sqrt{13}\\\\QS=\sqrt{(4-1)^2+(1-0)^2}=\sqrt{(3)^2+(1)^2}=\sqrt{9+1}=\sqrt{10}[/tex]

Since AB is congruent to QR, BC is congruent to RS, and AC is congruent to QS, this means that the two triangles are congruent.

Since ABC is congruent to DEF, and ABC is not congruent to JKL, this means that triangle DEF is not congruent to triangle JKL.

Since ABC is congruent to QRS, and QRS is not congruent to JKL, this means that triangle QRS is not congruent to JKL.

Since ABC is congruent to DEF and ABC is congruent to QRS, this means that DEF is congruent to QRS by the transitive property.

Answer:

The point [tex](\frac{\sqrt{3}}{2},-\frac{1}{2})[/tex]

Step-by-step explanation:

I added a graphic to the explanation.

Given the unit circle (the circle with radius equal to 1 unit centered at the point (0,0) ) we can represent its points only with an angle.

The point [tex]-(\frac{\pi}{6})[/tex] corresponds to the point that forms an angle of -30° respect to the positive axis-x (we measure the positive angles counterclockwise respect to the positive x-axis and the negative angles clockwise) because [tex]-(\frac{\pi}{6})[/tex] it is in radians and 180° = π radians ⇒

-(π/6) = - (180°/6) = - 30°

Given that we identify the point on the graph, we can find it coordinates using sine and cosine function :

[tex]sin(-30)=\frac{y1}{1} \\y1=-0.5=-\frac{1}{2}[/tex]

[tex]cos(-30)=\frac{x1}{1} \\x1=\frac{\sqrt{3}}{2}[/tex]

It is important to note that the hypotenuse of the right triangle which we used to apply sine and cosine is equal to 1 because is the radius of the unit circle.

The coordinates of the point are [tex](x1,y1)=(\frac{\sqrt{3}}{2},-\frac{1}{2})[/tex]

The point on the unit circle has the coordinates (√3/2, -1/2)

For a given angle a in an unit circle, the rectangular coordiantes of the point located in the circle are:

x = cos(a)

y = sin(a)

Here the angle is -(π/6), using the above relations, we will get the rectangular coordinates:

x = cos(-(π/6))

y = sin(-(π/6))

Simplify that:

x = √3/2

y = -1/2

The point is (√3/2, -1/2)

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Answer:

XY = 150

Step-by-step explanation:

AB = 8

W Z = 80

SF = 10

8 x 10 = 80

So if BC = 15

XY = 15 x 10 = 150

Answer:

Step-by-step explanation:

you can solve with a proportion

8 : 15 = 80 : x

x = 15 * 80 : 8

x = 150

or you find the rate from AD and WZ

8 : 80

simplify

1 : 10

15 * 10 = 150

No

The diagonal of the door opening is given by the Pythagorean theorem as ...

... √(96² +42²) = √8820 ≈ 94

Even if the table had zero thickness, it would not fit through the door

To determine if the circular table will fit through Jill's front door, we can use the Pythagorean theorem to compare the diagonal distance of the door to the diameter of the table.

To determine if the circular table will fit through Jill's front door, we can use the Pythagorean theorem. Assuming the door is rectangular, we can use the theorem to find the diagonal distance across the door and compare it to the diameter of the table. Let's calculate:

Let's plug in the values and calculate. If the diagonal distance is greater than or equal to 96 inches, then the table will fit through the front door.

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Answer:

[tex]\frac{BE}{EC} =\frac{1}{3}[/tex]

Step-by-step explanation:

In the diagram below we have

ABCD is a parallelogram. K is the point on diagonal BD, such that

[tex]\frac{BK}{CK} =\frac{1}{4}[/tex]

And AK meets BC at E

now in Δ AKD and Δ BKE

∠AKD =∠BKE                ( vertically opposite angles are equal)

since BC ║ AD and BD is transversal

∠ADK = ∠KBE     ( alternate interior angles are equal )

By angle angle (AA) similarity theorem

Δ ADK  and Δ EBK are similar

so we have

[tex]\frac{AD}{BE} =\frac{DK}{BK}[/tex]

[tex]\frac{AD}{BE} =\frac{4}{1}[/tex]

[tex]\frac{BC}{BE}=\frac{4}{1}[/tex]     ( ABCD is parallelogram so AD=BC)

[tex]\frac{BE+EC}{BE}=\frac{4}{1}[/tex]         ( BC= BE+EC)

[tex]\frac{BE}{BE} +\frac{EC}{BE}=\frac{4}{1}[/tex]

[tex]1+\frac{EC}{BE}=4[/tex]

[tex]\frac{EC}{BE}=3[/tex]  ( subtracting 1 from both side )

[tex]\frac{EC}{BE}=\frac{3}{1}[/tex]

taking reciprocal both side

[tex]\frac{BE}{EC} =\frac{1}{3}[/tex]

Answer:

∠1 = ∠2 = ∠3 = ∠4 = 28°

Step-by-step explanation:

A rhombus is a parallelogram with congruent sides. As with any parallelogram, the sum of adjacent interior angles is 180°. The figure is symmetrical, so either diagonal is also an angle bisector.

By any of various rules related to parallel lines and/or angle bisectors and/or isosceles trianges, all of the numbered angles are congruent (= α). Each of them is the complement of half the angle measure shown.

... α = (1/2)(180° -124°) = 90° -62° = 28°

The measure of the numbered angles in each rhombus is [tex]28^\circ[/tex] and this can be determined by using the properties of a rhombus.

Given :

A rhombus ABCD whose [tex]\rm \angle C = 124^\circ[/tex].

A rhombus is a quadrilateral whose all the sides are equal, opposite angles are equal, and opposite sides are parallel.

Line BD is the angle bisector and triangle BCD is the isosceles triangle and therefore, all the numbered angles 1, 2, 3, and 4 are equal.

[tex]\angle 1 = \angle 2 = \angle 3 = \angle 4[/tex]

[tex]\angle 1 = \angle 2 = \angle 3 = \angle 4 = \dfrac{1}{2}(180^\circ-124^\circ)[/tex]

[tex]\angle 1 = \angle 2 = \angle 3 = \angle 4 = \dfrac{1}{2}(56^\circ)[/tex]

[tex]\angle 1 = \angle 2 = \angle 3 = \angle 4 = (28^\circ)[/tex]

The measure of the numbered angles in each rhombus is [tex]28^\circ[/tex].

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Answer:

x=2

Step-by-step explanation:

3(3x+10)=50−x

The first step is to distribute the 3

3*3x + 3*10 = 50-x

9x+30 = 50-x

Add x to each side

9x+x+30 = 50-x+x

10x+30 = 50

Subtract 30 from each side

10x+30-30 = 50-30

10x= 20

Divide each side by 10

10x/10=20/10

x =2

Answer:

53 degrees

Step-by-step explanation:

Vertical angles are congruent so...

65x-12=43x+10

22x=22

x=1

Then add it into the equation of ∠A

m∠A= 65(1)-12

65-12

53

Vertical angles ∠A and ∠B are congruent, so their measures are set equal to each other to solve for x. Once x is found, it is substituted back into the expression for ∠A, resulting in a measure of 53 degrees for angle A.

Vertical angles are a pair of non-adjacent angles formed when two lines intersect. Since ∠A and ∠B are vertical angles, they are congruent, which means they have equal measures. We can set up an equation to solve for the variable x using the expressions for ∠A (65x − 12) and ∠B (43x + 10). Once x is found, we can substitute back into either expression to find the measure of ∠A in degrees.

To find the value of x, we set the expressions equal to each other: 65x − 12 = 43x + 10. Solving for x gives us: x = 22/22 = 1.

Now, substitute x back into the expression for ∠A: ∠A = 65(1) − 12 = 53°.

Therefore, ∠A measures 53 degrees.

b(3) = -16

b(2) = b(1) -7 = -2 -7 = -9

b(3) = b(2) -7 = -9 -7 = -16

Answer: b(3)= -16

Step-by-step explanation: