Write the equation of the line that passes through (2,3) and (4, 1) in point-slope form. Use (2,3) for (x1, y1).

36 cm

Let s represent the side of the square in cm. Then s+3 and s-2 are the sides of the rectangle of interest.

The area of the rectangle is the product of its side lengths:

... rectange area = (s+3)(s-2) = s² +s -6

The area of the square is the product of its side lengths, both of which are s.

... square area = s²

The difference of these areas is 30 cm², so ...

... rectangle area - square area = 30

... (s² +s -6) -(s²) = 30

... s = 36 . . . . . . . . . . . . simplify, add 6

The side of the square is 36 cm.

_____

Check

The rectangle dimensions are 39 cm by 34 cm, so its area is

... (39 × 34) cm² = 1326 cm²

The area of the square is (36 cm)² = 1296 cm²

The difference in areas is (1326 -1296) cm² = 30 cm², as required.

Answer:

5² - n  

Step-by-step explanation:

Five squared  = 5²

                     n =        n     Subtract n from 5²

                 Diff. = 5² - n

We indicate "taking the difference" by a "minus" sign, so all the other options are wrong.

Answer:

5² - n

Step-by-step explanation:

Five squared is written as = 5²

The symbol of n is n.

The term difference is subtraction.

The difference of five squared and n ⇒ 5² - n

Answer:

[tex]x=7.2\ units[/tex]

Step-by-step explanation:

we know that

In the right triangle of the figure

The tangent of angle of 67 degrees is equal to divide the opposite side to the angle of 67 degrees (17 units) by the adjacent side to angle of 67 degrees (x units)

[tex]tan(67\°)=17/x[/tex]

Solve for x

[tex]x=17/tan(67\°)[/tex]

[tex]x=7.2\ units[/tex]

Answer:

24 gallons

Step-by-step explanation:

18 divided by 3 is 6

6 x 4 = 24

so there are 24 gallons as a whole

You said . . . . . 18 = 3/4 g

Multiply each side by 4 . . . 72 = 3g

Divide each side by 3 . . . 24 = g

64

If each side length is multiplied by 8, the product of two side lengths will be multiplied by 8×8 = 64.

Area is proportional to the product of two side lengths, so will be multiplied by 64.

Answer:

a) [tex]-5x^{2}+3x+27[/tex]

b) [tex]-5x^{2}+3x+9[/tex]

Step-by-step explanation:

a) Let the required polynomial be p(x).

We have the relation, [tex]5x^{2}-3x-9[/tex] + p(x) = 18

i.e. p(x) = 18 [tex]-5x^{2}+3x+9[/tex]

i.e. p(x) = [tex]-5x^{2}+3x+27[/tex]

b) Let the required polynomial be q(x).

We have the relation, [tex]5x^{2}-3x-9[/tex] + q(x) = 0

i.e. q(x) = 0 [tex]-5x^{2}+3x+9[/tex]

i.e. q(x) = [tex]-5x^{2}+3x+9[/tex]

Answer:

(a) [tex]-5x^2+3x+27[/tex]

(b)  [tex]-5x^2+3x+9[/tex]

Step-by-step explanation:

(a)

Let the polynomial be Q(x).

Given polynomial P(x) = [tex]5x^2-3x-9[/tex]

As per the given statement: A polynomial(Q(x)) which, when added to the polynomial [tex]5x^2-3x-9[/tex], is equivalent to 18.

[tex]P(x)+Q(x) = 18[/tex]

[tex]5x^2-3x-9 +Q(x) = 18[/tex]

⇒[tex]Q(x) = 18 -(5x^2-3x-9)[/tex]

or

[tex]Q(x) = 18 -5x^2+3x+9[/tex]

Simplify:

[tex]Q(x) =-5x^2+3x+27[/tex]

Therefore, the polynomial is,  [tex]-5x^2+3x+27[/tex]

Check:

[tex]P(x)+Q(x)[/tex] = [tex]5x^2-3x-9 +(-5x^2+3x+27)[/tex]

                      = [tex]5x^2-3x-9 -5x^2 +3x+27[/tex]

                       = 18

(b)

Let the polynomial be Q(x).

Given polynomial P(x) = [tex]5x^2-3x-9[/tex]

As per the given statement: A polynomial(Q(x)) which, when added to the polynomial [tex]5x^2-3x-9[/tex], is equivalent to 0.

[tex]P(x)+Q(x) = 0[/tex]

[tex]P(x) = -Q(x)[/tex]

⇒[tex]Q(x) = -(5x^2-3x-9)[/tex]

or

[tex]Q(x) = -5x^2+3x+9[/tex]

Therefore, the polynomial is,  [tex]-5x^2+3x+9[/tex]

Check:

[tex]P(x)+Q(x)[/tex]=[tex]5x^2-3x-9 +(-5x^2+3x+9)[/tex]

                      = [tex]5x^2-3x-9-5x^2 +3x+9[/tex]

                       = 0

Answer:

none of the above

Step-by-step explanation:

The grocer's revenue will be the product of the number of loaves sold (30-2x) and their price (2.50+0.50x).

Revenue will be positive for values of x between those that make these factors be zero. The number of loaves sold will be zero when ...

... 30 -2x = 0

... 15 -x = 0 . . . . . divide by 2

... x = 15 . . . . . . . add x

The price of each loaf will be zero when ...

... 2.50 +0.50x = 0

... 5 + x = 0 . . . . . . . multiply by 2

... x = -5 . . . . . . . . . . subtract 5

Revenue will be positive for any number of increases greater than -5 and less than 15.

_____

D is the best of the offered choices, but it is incorrect in detail. -5 is a number less than 15, but will give zero revenue.

Answer:

x = −10 and x = 4

Step-by-step explanation:

x2 + 6x = 40

Subtract 40 from each side

x^2 + 6x -40 =0

Factor,  what 2 numbers multiply to -40 and add to 6

10 * -4 = -40    10+-4 = 6

(x+10) (x-4) = 0

Using the zero product property

x+10 =0     x-4=0

x=-10    x=4

Answer:

x = −10 and x = 4

Step-by-step explanation:

We are given the following quadratic equation and we are to solve it to find the two solution for the variable x:

[tex]x^2+6x=40[/tex]

Rearranging the equation by putting the constant on the same side as the variables to get:

[tex]x^{2} +6x-40=0[/tex]

Now factorizing it to get:

[tex]x^{2} -4x+10x-40=0\\\\x(x-4)+10(x-4)=0\\\\(x+10)(x-4)=0\\\\x= -10, x= 4[/tex]

Therefore, the solution to the given quadratic equation [tex]x^2+6x=40[/tex] are x = −10 and x = 4.

Answer:

A. 18.25

Step-by-step explanation:

After you do the multiplications, the problem is

... (56 +90)/8 = 146/8 = 18.25

_____

The division bar is a grouping symbol, equivalent to parentheses around both numerator and denominator. You must evaluate the numerator before you can divide by the denominator, which also must be evaluated before that division.

Answer:

Step-by-step explanation:

We know:

The sum of the measures of the angles of the triangle is equal to 180 °.

Therefore we have the equation:

[tex]y+(y-12)+56=180[/tex]         combine like terms

[tex]2y+44=180[/tex]              subtract 44 from both sides

[tex]2y=136[/tex]           divide both sides by 2

[tex]y=68[/tex]

Answer:

5.) 348 kg

6.) 20.7 lb

7.) 240

Step-by-step explanation:

5.) 48% of 725 kg

48 × 725 ÷ 100 = 348 kg

6.) 15% of 138 lb

15 × 138 ÷ 100 = 20.7 lb

7.) 45% of _____is 108.

100 × 108 ÷ 45 = 240

Thus, 5.) 348 kg; 6.) 20.7 lb; 7.) 240

-TheUnknownScientist

Answer:

[tex]\dfrac{mn^5}{p^4}[/tex]

Step-by-step explanation:

As with dividing any fractions, invert the denominator and multiply. Use the rules of exponents to combine factors.

[tex]\dfrac{\dfrac{m^2n^3}{p^3}}{\dfrac{mp}{n^2}}=\dfrac{m^2n^3}{p^3}\cdot\dfrac{n^2}{mp}\\\\=\dfrac{m^2n^3n^2}{p^3mp}=\dfrac{m^{2-1}n^{3+2}}{p^{3+1}}=\dfrac{mn^5}{p^4}[/tex]

_____

The applicable rules are ...

[tex]a^ba^c=a^{b+c}\\\\\dfrac{a^b}{a^c}=a^{b-c}[/tex]

After 5 years, the resale value of the laptop will be approximately $592.

The resale value of the laptop after each year can be calculated by multiplying the previous year's value by 0.75. Starting with the initial value of $2500, the resale value after 5 years can be found by multiplying $2500 by 0.75 five times. Let's calculate:

Therefore, the resale value of the laptop after 5 years will be approximately $592.

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Answer:

D -5/12

Step-by-step explanation:

2 1/2 = 2·(2/2) + 1/2 = 5/2

Dividing by 6 is the same as multiplying by 1/6.

... (-5/2)×(1/6) = -5·1/(2·6) = -5/12

If we let m represent the number of months, then the population increase of the first town is 100m and its decrease is 200m. The population decrease of the second town is 175 m.

We want to find m such that the increases and decreases make the towns' populations equal. We add the increases and subtract the decreases to the base population in each case.

... first town population = second town population

... 53075 -100m +200m = 55825 -175m . . . . . the model equation

Solution

... 100m = 2750 -175m . . . . . collect terms, subtract 53075

... 275m = 2750 . . . . . . . . . . add 175m

... m = 10 . . . . . . . . . . . . . . . . . divide by 275

The populations will be equal in 10 months.

Answer:

175m-125m+38,200=40,600-150m

Step-by-step explanation:

I just completed it on imagine math

f(x) = 8·1.25^x

An exponential function has the form ...

... f(x) = a·b^x

Then f(0) = a. Your graph shows f(0) = 8.

The base "b" can be found from any other point on the graph. You have marked the point (1, 10), so we can find "b" from ...

... f(1) = 10 = 8·b^1 = 8b

... 10/8 = b = 1.25 . . . . . . . divide by the coeffiicient of b

Now, we know the exponential function is ...

... f(x) = 8·1.25^x

Answer:

The correct answer is f(x) = 8·1.25^x

Step-by-step explanation:

(-1/4)x + 2 = 8

Multiply all terms by 4.

(-1)x + 8 = 32

Simplify.

-x + 8 = 32

Subtract 8 from both sides.

-x = 24

Multiply both sides by -1.

x = -24.

-1/4(-24) + 2 = 8

Evaluating functions is the process of substituting a specific value in place of the variable in a function to compute a result. For example, given the function f(x) = x^2, if asked to find f(2), you would replace x with 2 resulting in 4.

Evaluating functions in mathematics involves inserting a specific numerical value or a variable into a function to compute a result. For example, if you're given the function f(x) = x^2, and you're asked to find f(2), you simply substitute 2 in place of x in the function, resulting in 2^2, which equals 4.

For instance, if we were to evaluate f(5) for the given function, it would give the result 25 because 5^2 = 25. Hence, simply put, to evaluate a function is to replace the variable with the input value, and compute the result.

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For this case, we have that by definition:

Let "x" be an angle of any vertex of a right triangle.

[tex]tangent (x) = \frac {Cathet \ opposite} {Cathet \ adjacent}[/tex]

So, if we want to find the angle x of the triangle shown we have:

[tex]tangent (x) = \frac {13} {6}\\x = arc \ tangent (\frac {13} {6})\\x = 65.23[/tex]

Rounding:

[tex]x = 65\ degrees[/tex]

Answer:

65 degrees

Option d

Answer:

HL

Step-by-step explanation:

The two hypotenuses of these right triangles are marked congruent, and the leg QS is shared, hence congruent.

The HL theorem applies.

Answer:

x⁶ +30x⁵ +375x⁴ +2500x³ +9375x² +18750x +15625

Step-by-step explanation:

The expansion is the sum of C(6, k)·x^(6-k)·5^k for k=0–6, where ...

... C(6, k) = 6!/(k!(6-k)!)

For k = 0–6, C(6, k) = {1, 6, 15, 20, 15, 6, 1}

Then the expansion is ...

... x⁶ +6·5¹·x⁵ +15·5²·x⁴ +20·5³·x³ +15·5⁴·x² +6·5⁵·x +5⁶

Using binomial theorem to expand the power of a binomial (X+5) to the power 6 and the result is [tex]X^6 + 30 \times X^5 + 300 \times X^4 + 1500 \times X^3 + 3750 \times X^2 + 4500 \times X + 3125[/tex].

To expand the expression (X+5) to the power 6 using the binomial theorem, we can use the formula [tex](a + b)^n = nC_0 \times a^n + nC_1 \times a^{(n-1)} \times b^1 + nC_2 \times a^{(n-2)} \times b^2 + ... + nC_n * b^n[/tex].

In this case, a = X, b = 5, and n = 6.

Using the binomial coefficients, the expanded expression becomes:

[tex]X^6 + 6 \times X^5 \times 5 + 15 \timesX^4 \times 5^2 + 20 \times X^3 \times 5^3 + 15 \times X^2 \times 5^4 + 6 \times X \times 5^5 + 5^6[/tex]

Simplifying this expression gives

[tex]X^6 + 30 \times X^5 + 300 \times X^4 + 1500 \times X^3 + 3750 \times X^2 + 4500 \times X + 3125[/tex].

We know that , According to Pythagorean Theorem :

In a Right Angled Triangle :

✿  (Hypotenuse)² = (First Leg)² + (Second Leg)²

In the Figure : AC is the Hypotenuse and AB and BC are Two Legs

Given : Length of AB = 4 and Length of BC = 6

⇒ (AC)² = (AB)² + (BC)²

⇒ (AC)² = 4² + 6²

⇒ (AC)² = 16 + 36

⇒ (AC)² = 52

[tex]\mathsf{\implies AC = \sqrt{52}}[/tex]

[tex]\mathsf{\implies AC = 2\sqrt{13}}[/tex]

3rd Option is the Answer

Answer:

(x, y) = (5, 1)

Step-by-step explanation:

To eliminate x, you can double the second equation and subtract the first.

... 2(x +4y) -(2x -3y) = 2(9) -(7)

...11y = 11 . . . . . simplify

... y = 1 . . . . . . divide by 11

Using the second equation to find x, we have ...

... x + 4·1 = 9

... x = 5 . . . . . subtract 4

_____

Check

2·5 -3·1 = 10 -3 = 7 . . . . agrees with the first equation

(Since we used the second equation to find x, we know it will check.)

2.54 cm/in

In this context, "constant of proportionality" and "unit rate" mean the same thing.

Answer:

about 82%

Step-by-step explanation:

The distribution of sample means has a standard deviation that is the pipe standard deviation divided by the square root of the sample size. Thus, the standard deviation of the sample mean is 0.003/√9 = 0.001.

Then the limits on sample mean are 1.010 - 1×0.001 = 1.009 and 1.010 +2×0.001 = 1.012. The proportion of the normal distribution that lies between -1 and +2 standard deviations is about 81.9%.

The problem involves statistical calculation involving mean, standard deviation, and Z-scores of a normal distribution. We first calculate sample standard deviation, then the Z-scores for the given range. After finding probabilities for the Z-scores, we subtract to get the final probability of 0.8185.

The problem at hand involves the field of statistics, specifically, the normal distribution, sample mean, and standard deviation. We can use the following steps to solve the problem:

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Answer:

Step-by-step explanation:

riders : walkers = 10 : 1

The ratio tells you that 10 students ride the bus for every 1 student that walks. Since 10 is more than 1, more students ride the bus.

We know more students are riders, because we know that 10 is more than 1.

Answer:

The measure of angle B is positive

Step-by-step explanation:

we know that

Positive angles are those measured counterclockwise.

therefore

in this problem

The measure of angle B is positive

Answer:

Step-by-step explanation:

The circumference of a circle of radius r is given by

... C = 2πr

When r = 1 m, then

... C = 2π(1 m) = 2π m

The relation between time, distance, and speed is ...

... time = distance/speed

... time = (2π m)/(1 m/s) = 2π s

_____

Comment on the scenario

We have a hard time imagining what sort of scenario this is modeling, as it appears the "boat" is rotating in such a way as to place the barnacle above and below the water level. This problem may be nonsensical, but at least it is workable. (Some aren't.)

Answer:2 pi

Step-by-step explanation:

about 83%

Put the given value in the formula and do the arithmetic.

... P(66) = 90/(1 +271·e^(-0.122·66))

... = 90/(1 +271·e^-8.052)

... = 90/(1 +271·0.00031846)

... = 90/(1 +0.0863)

... = 90/1.0863

... = 82.8 . . . . percentage with some coronary heart disease