what is a piecewise function

Answer:

9. x=10

10. x=8

11. x=9/4=2.25

12. x=28

13. x=14

14. x=35

15. x=66

16. x=77/6

Step-by-step explanation:

9. x/20=18/36

Simplifying the fraction on the right side of the equation, dividing the numerator and denominator by 18:

x/20=(18/18) / (36/18)

x/20=1/2

Solving for x: Multiplying both sides of the equation by 20:

20(x/20)=20(1/2)

x=10

10. x/12=6/9

Simplifying the fraction on the right side of the equation, dividing the numerator and denominator by 3:

x/12=(6/3) / (9/3)

x/12=2/3

Solving for x: Multiplying both sides of the equation by 12:

12(x/12)=12(2/3)

x=4(2)

x=8

11. x/3=3/4

Solving for x: Multiplying both sides of the equation by 3:

3(x/3)=3(3/4)

x=9/4=2.25

12. x/42=30/45

Simplifying the fraction on the right side of the equation, dividing the numerator and denominator by 15:

x/42=(30/15) / (45/15)

x/42=2/3

Solving for x: Multiplying both sides of the equation by 42:

42(x/42)=42(2/3)

x=14(2)

x=28

13. x/12=14/12

Simplifying the fraction on the right side of the equation, dividing the numerator and denominator by 2:

x/12=(14/2) / (12/2)

x/12=7/6

Solving for x: Multiplying both sides of the equation by 12:

12(x/12)=12(7/6)

x=2(7)

x=14

14. x/25=28/20

Simplifying the fraction on the right side of the equation, dividing the numerator and denominator by 4:

x/25=(28/4) / (20/4)

x/25=7/5

Solving for x: Multiplying both sides of the equation by 25:

25(x/25)=25(7/5)

x=5(7)

x=35

15. x/110=(160-100)/100

x/110=60/100

Simplifying the fraction on the right side of the equation, dividing the numerator and denominator by 20:

x/110=(60/20) / (100/20)

x/110=3/5

Solving for x: Multiplying both sides of the equation by 110:

110(x/110)=110(3/5)

x=22(3)

x=66

16. x/11=(x+7)/17

Solving for x: Cross Multiplication:

17x=11(x+7)

Eliminating the parentheses on the right side of the equation:

17x=11(x)+11(7)

Multiplying:

17x=11x+77

Grouping the x's on the left side of the equation: Subtracting 11x both sides of the equation:

17x-11x=11x+77-11x

Subtracting:

6x=77

Dividing both sides of the equation by 6:

6x/6=77/6

Dividing:

x=77/6

Final answer:

To calculate the total distance traveled by the ball in the first four seconds, we sum the distances for each second, yielding 13k feet.

Explanation:

The student has provided a sequence of distances that a ball rolls down an inclined plane, with distances increasing each second. The distances in successive seconds are given by k feet, 2.5k feet, 4k feet, and so on. To calculate the total distance traveled by the ball in the first four seconds, we use the arithmetic sequence formula.

Let's calculate the distance for each second:

1st second: k feet

2nd second: 2.5k feet

3rd second: 4k feet

4th second: By the same pattern it follows that the 4th second should cover 5.5k feet.

To find the total distance, we sum these values:

Total distance = k + 2.5k + 4k + 5.5k = 13k feet.

This is the total distance traveled by the ball in the first four seconds.

The vertex of the parabola y = (x – 5)² + 8 with a vertical axis of symmetry is the point (5, 8).

The question is asking for the vertex of the parabola y = (x − 5)² + 8 which has a vertical axis of symmetry. In the general equation for a parabola, y - k = a(x - h)², the vertex is denoted by the (h, k). So, comparing this with the given parabola equation, we can determine that the h value corresponds to the value inside the brackets in the (x – h)² part of the equation, and the k value corresponds to the term added or subtracted outside the squared term.

For the given equation, y = (x – 5)² + 8, we can see that h = 5 and k = 8. Therefore, the vertex of the parabola is at the point (5, 8).

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The point-slope form of line:

[tex]y-y_1=m(x-x_1)[/tex]

The fomula of a slope:

[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

Substitute the coordinates of the points:

[tex]m=\dfrac{-4-2}{1-(-2)}=\dfrac{-6}{1+2}=\dfrac{-6}{3}=-2[/tex]

[tex]y-(-4)=-2(x-1)[/tex]       use distributive property

[tex]y+4=-2x+2[/tex]      subtract 4 from both sides

[tex]\boxed{y=-2x-2}[/tex]

Answer:

y = -2x-2

Step-by-step explanation:

The line passes through the points (-2, 2) and (1, -4).

The slope of the line is given by

[tex]m=\frac{y_2-y_1}{x_2-x_1}\\\\m=\frac{-4-2}{1+2}\\\\m=\frac{-6}{3}\\\\m=-2[/tex]

The slope intercept form of a line is given by

y = mx +b, where m is the slope and b is the y-intercept.

We have, m = -2. Hence, the equation is

y = -2x + b

Now, use the point (-2,2) to find b

2 = -2(-2) + b

2 = 4 +b

b = -2

Hence, the equation of the line is y = -2x-2

Third option is correct.

Answer:

16(3x - 1)

Step-by-step explanation:

the greatest common factor of 48 and 16 is 16, hence

48x - 16 = 16(3x - 1)

Final answer:

After factoring out the greatest common factor of 16 from the expression 48x-16, the correct factored form is 16(3x - 1). The friend's factorization is correct if this result was obtained, demonstrating the universal application of mathematical rules.

Explanation:

To evaluate if your friend factored the expression 48x-16 correctly, we look for the greatest common factor that both terms in the expression share. Both 48x and 16 are divisible by 16. Factoring out the greatest common factor of 16, we get:

16(3x - 1)

This is the factored form of the expression 48x-16, which shows that your friend is correct if they arrived at this result. This method is an application of the distributive property, which states that a(b + c) = ab + ac. In this case, we take out a '16' from both terms, which is the inverse process - factoring.

Moreover, the principles that apply to factoring expressions are the same regardless of location or time, demonstrating the universality of mathematical rules. Whether we are factoring polynomials or checking arithmetic, using the correct rule and applying it properly is essential.

Answer:

A

Step-by-step explanation:

given y = 3(x + 2)(x - 2) = 3(x² - 4) = 3x² - 12

for the y- intercept let x = 0

f(0) = - 12 → y-intercept is the point (0, - 12) → True

for x-intercepts let y = 0

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

x + 2 = 0 ⇒ x = - 2

x - 2 = 0 ⇒ x = 2

x- intercepts are (- 2, 0) and (2, 0) → not (0, - 12)

since f(x) = 3x² - 12 is symmetric about the y-axis

the vertex is at (0, - 12 ) → not (- 2, 2)

Answer:

value of QZ = 8 units and QM =  12 units.

Step-by-step explanation:

Given: In triangle PQR has medians QM and PN that intersect at Z.

If ZM = 4 units.

In the figure given below;  second median divided the two triangles formed by the first median in the ratio 2:1.

We have to find the value of QZ and QM;

QZ:ZM = 2: 1

⇒ [tex]\frac{QZ}{ZM} = \frac{2}{1}[/tex]

Substitute the value of ZM =4 units and solve for QZ;

[tex]\frac{QZ}{4} = \frac{2}{1}[/tex]

Multiply both sides by 4 we get;

[tex]QZ = 2 \times 4 = 8 units[/tex]

Now, calculate QM;

QM = QZ+ZM = 8 + 4 = 12 units.

Therefore, the value of QZ and QM are; 8 units and 12 units

Answer:

221.0001

Step-by-step explanation:

Attempt to look at the number of 0s behind the decimal point, which you add those 0s and the digit number (1) into place

221

  0.0001

221.0001

Answer:

300 dollars of interest

Step-by-step explanation:

5000 (0.06)

300

Answer:

$30

Step-by-step explanation:

You are being asked to figure out simple interest, I think.

Givens

Formula

i = P*r*t

Solution

Edit

You want this to be done using a % proportion. Is that correct from what I read in the comments after the question (which I didn't see -- sorry).

So it comes to the same thing.

The x-intercept is for y = 0.

substitute y = 0 to the equation of the function:

[tex]\dfrac{1}{3}x+y=-15\\\\\dfrac{1}{3}x+0=-15\\\\\dfrac{1}{3}x=-15\qquad\text{multiply both sides by 3}\\\\\boxed{x=-45}[/tex]

Answer:

(-45,0)

Step-by-step explanation:

I took the test

Answer:

38.94

Step-by-step explanation:

To find the area, multiply the length times the width. In this case 9.5 times 4.1.

Answer:

t > - 27

Step-by-step explanation:

t + 4 > 31.  Next, solve for t:  t > -27

Answer:

t+4 > 31

t > 27

Step-by-step explanation:

(t + 4) exceeds 31

t+4 > 31

Subtract 4 from each side

t+4-4 > 31-4

t > 27

Answer:

640 bacteria.

Step-by-step explanation:

Make a chart.

Bacteria    Weeks

40                  0

80                  1

160                  2

320                 3

640                 4

It starts off with 40 bacteria, which means no weeks have passed so far. That is why 40 pairs with 0 rather than 1.

Answer:

5 years.

Step-by-step explanation:

She earns $339.3 per year on interest rate. So I divided 1,696.50 by 339.3 to get the amount of ears.

Lina invested for 5 years.

According to the problem,

So, we can say $ 1,696.50 is the interest

∴ Prt = 1696.50

⇒ (7800)(4.35/100)t = 1696.50

⇒ t = 5

∴ Line invested for 5 years.

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Answer:

60 pounds

-hope this helps

Answer:

£60

Step-by-step explanation:

100% represents the original price of the shoes

then (100 - 10) = 90% of the original price

Divide the cost by 90% to find 1% then multiply by 100 for the original price

original price = [tex]\frac{54}{90}[/tex] × 100 = £60

Triangles NPM and ABC are similar by SAS, establishing that angle ABC corresponds to angle NPM. Thus, <ABC = <MNP.

In triangles NPM and ABC, the given information suggests that these triangles are similar by the Side-Angle-Side (SAS) criterion. Specifically, MP = AC, AB = NM, and BC = PN, along with the corresponding equal angles A = M, B = N, and C = P.

The Angle-Angle Similarity Postulate asserts that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Therefore, triangles NPM and ABC are similar.

The corresponding angles in similar triangles are equal. Hence, angle ABC is equal to angle NPM. Therefore, the correct answer is:

<MNP

This is because angle ABC in triangle ABC corresponds to angle NPM in triangle NPM.

Answer: 7/16 or .4375

Step-by-step explanation: 1/4 of 1 is 1/4. 1/4 of 3/4 is .1875 which also equals 3/16. Add 1/4 and 3/16 to get 7/16 cups of sugar or 0.4375. Hope this helps!

Answer:

(3 + 5i) – (2 + i)

Step-by-step explanation:

(–2 + 6i) – (1 – 2i)

The real parts  -2-1 = -3

The imaginary parts 6i--2i = 6i+2i = 8i

(–2 + 6i) – (–1 – 2i)

The real parts -2 +1 = -1

The imaginary parts 6i --2i = 6i+2i =8i

(3 + 5i) – (2 – i)

The real parts 3-2 =1

The imaginary parts = 5i --i = 5i+i = 6i

(3 + 5i) – (2 + i)

The real parts 3-2 =1

The imaginary parts 5i -i = 4i

Answer:

d.(3 + 5i) – (2 + i)

Step-by-step explanation:

Answer:

y = 4.5 x

Step-by-step explanation:

If this is a proportional relationship, we can find the slope

m = (y2-y1)/(x2-x1)

m = (27-0)/(6-0)

m = 27/6

m=9/2

The slope is 9/2

The y intercept, where it crosses the y axis is 0

Using the slope intercept form of the equation y=mx+b

y = 9/2 x+0

y =9/2x

It wants the decimal form

y = 4.5 x

Answer:

tn = a*2^(n - 1)

Step-by-step explanation:

a = 4

r = 2

tn= a*2^(n-1)

Try it

t3 = 4*2^(3 -1)        Combine the power

t3 = 4 * 2^2            Find 2^2

t3 = 4 * 4                Multiply

t3 = 16                     Answer for t3

Answer:

2^(n+1)

Step-by-step explanation:

You don't even need to use algebraic methods or the formula for this; it's just common sense. 4=2^2, 8=2^3, 16=2^4.

4 is the term 1

8 is term 2

2=term # (1) + 1 for 4

3 = term # (2) + 1 for 8

And so on

So it is 2 to the power of (term # + 1)

Look at the picture.

Use the Pythagorean theorem to solve h:

[tex]h^2+5^2=7^2[/tex]

[tex]h^2+25=49[/tex]            subtract 25 from both sides

[tex]h^2=24\to h=\sqrt{24}[/tex]

ΔADC and ΔCDB are similar. Therefore the sides are in proportion:

[tex]\dfrac{h}{5}=\dfrac{x}{h}[/tex]       cross multiply

[tex]5x=h^2[/tex]      put the value of h

[tex]5x=(\sqrt{24})^2[/tex]

[tex]5x=24[/tex]       divide both sides by 5

[tex]x=4.8[/tex]

y = 5 + x → y = 5 + 4.8 = 9.8

Or other method. ΔADC and ΔACB are similar. Therefore the sides are in proportion:

[tex]\dfrac{y}{7}=\dfrac{7}{5}[/tex]        multiply both sides by 7

[tex]y=\dfrac{49}{5}\\\\\boxed{y=9.8}[/tex]

Answer:

C

Step-by-step explanation:

$4.50 x 4 = $18.00          $6.00 x 2 = $12.00          $18 + $12  = $30

$4.50 + $6.00 = $10.50

Answer:

5f

Step-by-step explanation:

-5+x=0

Add 5 to both sides of the equation.

x=5

Answer:

We need to add 5F to make the sum 0

Step-by-step explanation:

-5 +  temperature = 0

Add 5 to each side

-5 + 5 +  temperature=0 + 5

temperature = 5

We need to add 5F to make the sum 0

Answer:

[tex]y=\frac{3w}{10\left(2w-1\right)}[/tex]

Step-by-step explanation:

We are given

Miss Teng normal walk:

speed =0.5 m/s

distance traveled =y km =y*1000

now, we can find time

we know

time = distance / speed

[tex]t_1=\frac{1000y}{0.5}[/tex]

[tex]t_1=2000y[/tex]

Miss Teng  walk on cloudy day:

speed =w m/s

distance traveled =y km =y*1000

now, we can find time

we know

time = distance / speed

[tex]t_2=\frac{1000y}{w}[/tex]

If she reached her school 5 minutes earlier than usual

so,

[tex]t_1-t_2=5\times 60 sec[/tex]

[tex]t_1-t_2=300[/tex]

now, we can plug values

[tex]2000y-\frac{1000y}{w}=300[/tex]

now, we can solve for y

we can multiply both sides by w

[tex]2000yw-\frac{1000y}{w}w=300w[/tex]

[tex]2000wy-1000y=300w[/tex]

[tex]1000y\left(2w-1\right)=300w[/tex]

[tex]y=\frac{3w}{10\left(2w-1\right)}[/tex]

Answer:

Step-by-step explanation:

We can set up an equation:

4(3x+4)=76

We can distribute the left side and get:

12x+16=76

Subtract 16 from both sides:

12x=60

Divide 12 from both sides:

x=5

Check:

4(3*5+4)

4(15+4)

4(18)

76

So x=5

Answer:363

Step-by-step explanation:

We have to find partial sum for the sequence 243 , 81 , 27 ..... up to 5 terms(S5 given)

The sequence actually is 3^5,3^4.,3^3.....

Therefore first 5 terms are 1) 3^5 I.e. 243

2) 3^4 I.e. 81

3) 3^3 I.e. 27

4)) 3^2 I.e. 9

5) 3^1 I.e. 3

Adding all those no. we get partial sum of first 5 no. of the sequence

So, 243 + 81 + 27 + 9 + 3

= 363

Hope it helps!!!

Answer:

D. 363 is the one

Answer:

119

Step-by-step explanation:

381-262

Answer:

x^3 +2x^2 -4x -8

Step-by-step explanation:

The factors of x^2 +4x +4 are (x +2)^2.

The factors of 4 -x^2 are -1(x -2)(x +2).

The least common denominator is the least common multiple of these, excluding the -1 multiplier. That multiple is ...

... (x +2)^2 × (x -2) = x^3 +2x^2 -4x -8

_____

Comment on the minus sign

When the given fractions are expressed using the denominator above, the second one will need to have its numerator sign adjusted. The two fractions are ...

The LCD or Least Common Denominator for the fractions 7/(x^2 + 4x + 4) and 5/(4 - x^2) is the product of the distinct denominators after factoring, which is (x+2)^2(4 - x^2), given they have no common factors.

The acronym LCD refers to the Least Common Denominator, which in mathematics, is the least common multiple of the denominators of a set of fractions. It is used to find a common denominator so that the fractions can be combined or compared. For the fractions 7/x^2 + 4x + 4 and 5/4 - x^2, the LCD would be the product of the distinct denominators assuming they cannot be factored into common factors. In this case, assuming x^2 + 4x + 4 cannot be further factored (it is a perfect square: (x+2)^2), the denominators are x^2 + 4x + 4 and 4 - x^2 (since we consider x^2 to have a negative coefficient here). Consequently, if there are no common factors, the LCD is (x^2 + 4x + 4)(4 - x^2) or (x+2)^2(4 - x^2).

Answer:

2A /b = h

Step-by-step explanation:

A=1/2 bh

We want to isolate h

First we multiply by 2

2A = 2 * 1/2 bh

2A = bh

Then we divide by b to isolate h

2A /b = bh/b

2A /b = h

Answer:

8

Step-by-step explanation:

Substitute the equation and values into the average rate of change formula.