Mila reads at a rate of 2 paragraphs per minute. After reading for 3 minutes, she had read a total of 6 paragraphs. This situation can be represented with a linear equation written in point-slope form, where x represents the number of minutes and y represents the number of paragraphs. Use this information to complete each statement about the linear equation.

Answer:

f(x)=x-1

Step-by-step explanation:

replace y by f(x) to obtain functional notation

f(x) = x - 1

Answer:

Correct options are A and D

Step-by-step explanation:

According to the synthetic division in the diagram you can write down the result of division:

[tex]2x^2+9x-7=(x-(-6))(2x-3)+11,\\ \\2x^2+9x-7=(x+6)(2x-3)+11.[/tex]

Therefore,

When [tex]x= - 6,2{x^2} + 9x -7= 11[/tex] and when [tex]2{x^2} + 9x -7[/tex] is divided by [tex]\left( {x + 6} \right)[/tex], the remainder is 11.Option (A) is correct and option (D) is correct.

Further Explanation:

Given:

Explanation:

The synthetic division can be expressed as follows,

[tex]\begin{aligned}- 6\left| \!{\nderline {\,{2\,\,\,\,\,\,\,\,\,\,9\,\,\,\,\,\,\,\,\,\,\, - 7} \,}} \right.  \hfill\\\,\,\,\,\,\,\underline {\,\,\,\,\,\,\,\, - 12\,\,\,\,\,\,\,\,\,\,\,\,18}  \hfill\\\,\,\,\,\,\,\,\,2\,\,\,\,\,\, - 3\,\,\,\,\,\,\,\,\,\,\,\,11 \hfill \\\end{aligned}[/tex]

The last entry of the synthetic division tells us about remainder and the last entry of the synthetic division is 11. Therefore, the remainder of the synthetic division is 11.

When [tex]x= - 6, 2{x^2} + 9x -7= 11[/tex] and when [tex]2{x^2} + 9x -7[/tex] is divided by [tex]\left( {x + 6} \right)[/tex], the remainder is 11. Option (A) is correct and option (D) is correct.

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Answer details:

Grade: High School

Subject: Mathematics

Chapter: Synthetic Division

Keywords: division, factor (x+5), remainder 12, statements, true, apply, divided, binomial synthetic division, long division method, coefficients, quotients, remainders, numerator, denominator, polynomial, zeros, degree.

Yes this is a Law of Detachment.

If you need to, you can write and solve an equation for the factor you seek.

... 6×10^10 = factor × 2×10^-3

Divide by 2×10^-3 to find the value of the factor:

... (6×10^10)/(2×10^-3) = factor

... factor = (6/2)×10^(10-(-3))

... factor = 3×10^13

The first number is 3×10^13 times the second number.

_____

An exponent signifies repeated multiplication.

... 10×10×10 = 10³

Just as you cancel common factors when you do division, you can subtract exponents.

[tex]\dfrac{10\cdot 10\cdot 10}{10\cdot 10}=\dfrac{10}{1}=10\\\\\dfrac{10^3}{10^2}=10^{3-2}=10^1=10[/tex]

The same process works regardless of the signs of the exponents. When multiplying, we add exponents; when dividing we subtract the exponent of the denominator.

y = [tex]\frac{2}{5}[/tex] x + [tex]\frac{27}{5}[/tex]

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y-intercept )

y = [tex]\frac{2}{5}[/tex] x - 6 is in this form with slope m = [tex]\frac{2}{5}[/tex]

Parallel lines have equal slopes, thus

y = [tex]\frac{2}{5}[/tex] x + c is the partial equation of parallel line

to find c , substitute (- 1, 5 ) into the partial equation

5 = - [tex]\frac{2}{5}[/tex] + c ⇒ c = 5 + [tex]\frac{2}{5}[/tex] = [tex]\frac{27}{5}[/tex]

y = [tex]\frac{2}{5}[/tex] x + [tex]\frac{27}{5}[/tex] ← equation of parallel line

The point-slope form of a line:

[tex]y-y_1=m(x-x_1)\\\\m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

We have the points (5, 6) and (3, 4). Substitute:

[tex]m=\dfrac{4-6}{3-5}=\dfrac{-2}{-2}=1\\\\y-6=1(x-5)\\\\y-6=x-5\qquad|\text{add 6 to both sides}\\\\y=x+1\qquad|\text{subtract x from both sides}\\\\-x+y=1\qquad|\text{change the signs}\\\\x-y=-1[/tex]

Answer:

slope-intercept form: y = x + 1

point-slope form: y - 6 = 1(x - 5)

standard form: x - y = -1

Hey there!

Given points:

...(5,6) and (3,4)

Slope-intercept form:

... y=mx+b

'm' is the slope and 'b' is the y-intercept.

Slope:

... (y₂-y₁)/(x₂-x₁)

... (4-6)/(3-5)

... -2/-2

...1

:

... y = x + b

... 4 = 3 + b

... b = 1

Slope-intercept form:

... y = x + 1

Hope helps!

Answer:

Step-by-step explanation:

Apparently, your periodic rate is that used to compute the interest accrued each period. It seems to be the compound (annual) rate divided by the number of periods in a year: quarterly, 4; monthly, 12; daily, 365; biweekly, 26; semimonthly, 24.

_____

If you want the effective annual rate to be 18% in each case, the numbers are different. For n periods per year, those are calculated as

[tex]\sqrt[n]{1.18}-1[/tex]

A periodic rate of 0.04224664 will give an effective annual rate of 18%.

Periodic rates for a 18% compound rate compounded quarterly, monthly, daily, biweekly, and semimonthly are calculated as follows :

Using Pythagorean's Theorem we know that a^2 + b^2 = c^2

C is the length of the ladder, and we are given one of the sides, let's call that side b

                                                                                          _________

we have a^2 + b^2 = c^2, and a^2 = c^2 - b^2, so a = √ c^2 - b^2

       _________      ______     ____

a = √12^2-3^2 = √ 144-9 = √ 135 = 11.61895 

so the top of the ladder is 11.6 feet above the ground

here u go hope this helps

Answer:

12 sin60°

Remember SOHCAHTOA.

sinθ =

opposite

hypotenuse

sin60° =

x

12

x = 12 sin60°

f(x) = - [tex]\frac{6}{7}[/tex] x + [tex]\frac{53}{7}[/tex]

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y-intercept )

here m = - [tex]\frac{6}{7}[/tex]

partial equation is y = - [tex]\frac{6}{7}[/tex] x + c

to find c substitute (3, 5 ) into the partial equation

5 = - [tex]\frac{18}{7}[/tex] + c ⇒ c = [tex]\frac{53}{7}[/tex]

f(x) = - [tex]\frac{6}{7}[/tex] x + [tex]\frac{53}{7}[/tex]

1. The vertical asymptote requires the denominator have a zero at that location. The x-intercept requires the numerator have a zero at that location. The horizontal asymptote amounts to a multiplier of the function:

... y = 2(x +5)/(x -3)

2. The vertical asymptote requires the denominator have a zero at that location. The oblique asymptote is an add-on

... y = 1/(x +1) +(x +2)

... y = (x² +3x +3)/(x +1)

We can use rational functions to define functions with specific characteristics. A function with a vertical asymptote at x = 3, a horizontal asymptote at y = 2, and x-intercept at x = -5 could be written as f(x) = 2(x + 5) / (x - 3). A function with a vertical asymptote at x = -1 and an oblique asymptote at y = x + 2 can be written as f(x) = (x^2 + x - 2) / (x + 1).

The subject here pertains to certain characteristics of functions, specifically regarding asymptotes and intercepts. In order to create a function with the required characteristics, you would typically use rational functions.

Vertical asymptotes occur when the denominator of a function is zero, horizontal asymptotes are connected to the degree of the polynomials in the function, and x-intercepts occur when the function itself equals zero.

Here's how we can write the function for each case:

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(A) f(a + 2) = a² - 16

substitute x = a + 2 into f(x)

f(a + 2) = (a + 2)² - 4(a + 2) - 12

           = a² + 4a + 4 - 4a - 8 - 12

           = a² - 16

(B ) f(a + h) = a² + 2ah + h² - 4a - 4h - 12

substitute x = a + h into f(x)

f(a + h) = a² + 2ah + h² - 4a - 4h - 12

Y = 1/2 + 5 , the slope of this equation is 1/2

The equations of the parallel lines must also have a slope of 1/2.

This is because parallel lines have the same value of slope.

point slope form

y-y1 = m (x-x1)

y- (-2) = 3(x-1)

y+2 = 3(x-1)

y + 2 = 3(x - 1)

the equation of a line in point-slope form is

y - b = m(x - a)

where m is the slope and (a, b) a point on the line

here m = 3 and (a, b) = (1, - 2), hence

y + 2 = 3(x- 1) ← in point-slope form

The percent change is given by ...

... (percent change) = (new amount - old amount)/(old amount) × 100%

This can be rearranged to give a formula for the new amount. First, we'll rewrite it to a slightly different form.

... (percent change) = ((new amount)/(old amount) -1) × 100%

... (percent change)/100% = (new amount)/(old amount) -1 . . . . divide by 100%

... (percent change)/100% + 1 = (new amount)/(old amount) . . . add 1

... (old amount) × ((percent change)/100%) +1) = new amount . . . . multiply by old amount

We can now use this formula to find the new amount in each case.

13. 25 × (300%/100% +1) = 25 × 4 = 100 . . . . dollars

14. 160 × (-20%/100% +1) = 160 × 0.8 = 128 . . . . bananas

15. 56 × (-75%/100% +1) = 56 × .25 = 14 . . . . books

16. 52 × (25%/100% +1) = 52 × 1.25 = 65 . . . . companies

17. 12000 × (5%/100% +1) = 12000 × 1.05 = 12,600 . . . . miles

18. 710 × (-10%/100% +1) = 710 × 0.90 = 639 . . . . points

_____

Considering the above formula for percent change (or its "slightly different form"), you may want to reconsider your answers for problems 7–12.

ANSWER

The solution is

[tex](x=1,y=2),(x=2,y=3)[/tex]

EXPLANATION

We have

[tex]y=x+1---(1)[/tex]

and

[tex]y=x^2-1---(2)[/tex]

Let us substitute equation (1) in to equation (2). This gives us,

[tex]x+1=x^2-1(2)[/tex]

We rewrite this as a quadratic equation as the highest degree is 2.

[tex]x^2-x-1-1=0[/tex]

This implies that

[tex]x^2-x-2=0[/tex]

we factor to obtain,

[tex]x^2+x-2x-2=0[/tex]

[tex]x(x-1)-2(x-1)=0[/tex]

[tex](x-1)(x-2)=0[/tex]

This means,

[tex](x-1)=0\:\: or\:\:(x-2)=0[/tex]

[tex]x=1\:\: or\:\:x=2[/tex]

We substitute this values into any of the above equations, preferably equation (1)

When, [tex]x=1[/tex], [tex]y=1+1=2[/tex]

When, [tex]x=2[/tex], [tex]y=2+1=3[/tex]

The solution is

[tex](1,2),(2,3)[/tex]

Replace the variables with their values and do the arithmetic.

qxs = -30 +2(600) -4(60) = -30 +1200 -240

qxs = 930

930 of product x is produced.

Explanation of how to determine the quantity of product x produced when given specific prices, the quantity produced of product x is -150.

Supply Function: qxs = - 30 + 2px - 4pz

a. To determine the quantity produced when px = $600 and pz = $60, substitute these values into the supply function:

qxs = -30 + 2(600) - 4(60) = -30 + 120 - 240 = -150

Therefore, when px = $600 and pz = $60, the quantity produced of product x is -150.

A.) If Grace and Claire’s parents each invested $7,600 into a college savings account when the girls were born, how much money will each girl have for college when she turns 18? Explain.

Answer/Explanation:

For Grace, I plugged in 18 for x.

Work:

y= 1000(18) + 7600

y= 18000 = 7600

y= 25600

For Claire, I had to first find the slop by using rise over run.

11600-10000/3= 800

So the formula for Claire is y= 800x + 7600 and then you just plug in 18 for x.

Work:

y= 800(18) + 7600

y= 14,400 + 7600

y= 22,000

Therefore, Grace would have $25,600 at the end of 18 years old and Claire would have $22,000 when she turns 18 years old.

                 I hope that helped you or anyone else!! :)

The amount of money that Grace and Claire will have for college when they turn 18 will be $25600 and $22000 respectively.

Based on the information given, the amount that Grace will make will be:

= 1000x + 7600

= 1000(18) + 7600

= 18000 + 7600

= 25600.

The amount that Claire will make will be:

= 800x + 7600.

= 14400 + 7600

= 22000

The functions show a positive correlation between time and the amount of money saved.

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Expressed as a fraction 56 is 56/35 of 35. That fraction can be reduced, and expressed several ways.

56/35 = 8/5 = 1 3/5 = 1.6

56 is 1 3/5 of 35

56 is 1.6 times 35

We can let x and y represent cups of lemonade and numbers of lemon bars, respectively. Then the constraints are ...

A graph is shown in the attachment, with the ordered pairs plotted. It is not feasible to sell half cups of lemonade or half lemon bars, so the second and third choices must be excluded. The point (160, 110) falls outside the feasible region, so is not a correct choice.

The correct choices are ...

Final answer:

The only ordered pairs that meet the constraints of Larry's Lemons business situation are (150,200) and (180,100), as these combinations satisfy the conditions for revenue, expenses, and minimum amount of lemonade sold.

Explanation:

To find which ordered pairs (x, y) make sense in the context of Larry's Lemons' business situation, we need to consider the given conditions and set up inequalities to express the sales and expense constraints.

The conditions are:

Revenue Constraint:

For the revenue, we need to ensure that the seller made at least $500.

2x + 1.5y ≥ 500

Expense Constraint:

For the expenses, they should not exceed $100.

0.25x + 0.20y ≤ 100

Lemonade Sold Constraint:

Since at least 150 cups of lemonade were sold:

x ≥ 150

Now we check the given ordered pairs against these constraints:

Only the ordered pairs (150,200) and (180,100) satisfy all the conditions of the problem.

Try this option:

x²+27=0;

[tex](x+\sqrt{-27})(x- \sqrt{-27})=0; \ => \ \left[\begin{array}{ccc}x=3 \sqrt{3}i\\x=-3 \sqrt{3}i \end{array}\right[/tex]

x = ± 3i√3

given x² + 27 = 0 (subtract 27 from both sides )

x² = - 27 ( take the square root of both sides )

x = ±[tex]\sqrt{-27}[/tex] = ± √(9 × 3 × -1 ) ← (i = √-1 )

  = ± (√9 × √3  ×√-1 ) = ±3i√3

The answer would be 56 because 14X4 = 56

There were 56 girls who tried out for the volleyball team, found by dividing the number of girls on the team (14) by the percentage that made the team (25%), which equals 14 divided by 0.25.

The question asks us to find the total number of girls who tried out for the volleyball team if 14 girls (making up 25% of those who tried) made the team. To calculate the total number of girls who tried out, we can set up the equation based on the percentage:

25% of total girls = 14

We can rewrite 25% as 0.25 in decimal form:

0.25 × total girls = 14

To find the total number of girls, we divide both sides of the equation by 0.25:

total girls = 14 ÷ 0.25

total girls = 56

Therefore, 56 girls tried out for the volleyball team.

(1)

take out the common factor (y - 4 )

= (y - 4)(3x - 2)

(2)

factor by grouping (1/2 terms and 3/4 terms )

4y(5x - 1) + 7(5x - 1)

take out the common factor (5x - 1)

= (5x - 1)(4y + 7)

1.  Factor out the (y-4)

(y-4) (3x-2)

2.Factor by grouping

20xy -4y    +35x -7

4y(5x-1) +7(5x-1)

then factor out 5x-1

(5x-1)(4y+7)

Since the triangle is equilateral (you can tell by the tick on each side), all angles have the same measure as well.

The angles of a triangle sum up to 180 degress, so three equal angles must measure 60 degrees each.

So, in particular, we have

[tex] 7x+4 = 60 \iff 7x = 56 \iff x=8[/tex]

x = 8 and y = 6

ΔRST is an equilateral triangle

with all 3 sides equal in length and all 3 angles = 60°, hence

7x + 4 = 60 ( subtract 4 from both sides )

7x = 56 ( divide both sides by 7 )

x = 8

similarly

8y + 12 = 60 ( subtract 12 from both sides )

8y = 48 ( divide both sides by 8 )

y = 6

Try this solution (all the details are in the attached picture, answers are underlined with colour).

The perimeter of triangle of ABC is [tex]\boxed{28.73}[/tex] and the area of triangle ABC is [tex]\boxed{37.86}.[/tex]

Further explanation:

Given:

The measure of angle A is [tex]\angle A = {60^ \circ }.[/tex]

The measure of angle C is [tex]\angle C = {45^ \circ }.[/tex]

The length of side AB is [tex]AB = 8[/tex]

Calculation:

The sum of all angles of a triangle is [tex]{180^ \circ }.[/tex]

[tex]\begin{aligned}\angle A + \angle B + \angle C &= {180^ \circ }\\{60^ \circ } + \angle B + {45^ \circ } &= {180^ \circ }\\{105^ \circ } + \angle B &= {180^ \circ }\\\angle B&= {180^ \circ } - {105^ \circ }\\\angle B&= {75^ \circ }\\\end{aligned}[/tex]

The sine rule in triangle ABC can be expressed as,

[tex]\begin{aligned}\frac{{BC}}{{\sin {{60}^ \circ }}}&=\frac{8}{{\sin {{45}^ \circ }}}\\BC&=\frac{8}{{\frac{1}{{\sqrt2 }}}}\times\frac{{\sqrt 3 }}{2}\\BC&= 9.80\\\end{aligned}[/tex]

The length of AC can be calculated as follows,

[tex]\begin{aligned}\frac{{AB}}{{\sin {{45}^ \circ }}} &= \frac{{AC}}{{\sin {{75}^ \circ }}}\\\frac{8}{{\sin {{45}^ \circ }}} \times \sin {75^ \circ }&= AC\\10.93 &= AC\\\end{aligned}[/tex]

The perimeter of triangle ABC can be obtained as follows,

[tex]\begin{aligned}{\text{Perimeter}}&= AB + BC + AC\\&= 8 + 9.80 + 10.93\\&= 28.73\\\end{aligned}[/tex]

The area of triangle ABC can be obtained as follows,

[tex]\begin{aligned}{\text{Area}}&=\frac{1}{2} \times AB \times AC \times \sin \left( A \right)\\&= \frac{1}{2} \times 8 \times 10.93 \times \sin {60^ \circ }\\&= 4 \times 10.93 \times \frac{{\sqrt3 }}{2}\\&= 37.86\\\end{aligned}[/tex]

The perimeter of triangle of ABC is [tex]\boxed{28.73}[/tex] and the area of triangle ABC is [tex]\boxed{37.86}.[/tex]

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Answer details:

Grade: Middle School

Subject: Mathematics

Chapter: Triangles

Keywords: angles, ABC, angle A=60 degree, perimeter, area of triangle, triangle ABC.

You will need to find the area of the rectangular shaped wooden floor and divide that area by 115 square feet.

46 x 35 = 1610

1610 square feet/115 square feet

= 14 cans of wood stain

Answer:

It should be 14

Step-by-step explanation: