Complete the square to determine the minimum or maximum value of the function defined by the expression.

x2 − 12x − 2

A) maximum value at 38 B) minimum value at 38 C) maximum value at −38 D) minimum value at −38

Answer:

yesterday 9 today 10

Step-by-step explanation:

46/5= 9.2

52/5= 10.4

Explanation:

The denominator of "miles per hour" is 1 hour, a single unit. When the denominator of a rate is 1 unit, it is called a unit rate.

Answer:

3-,3,0,3

Step-by-step explanation:

For this case we must evaluate the following quadratic equation, [tex]y = -x ^ 2 + 3,[/tex] for [tex]x = -3[/tex], [tex]x = 0[/tex]and [tex]x = 3[/tex]

For [tex]x = -3[/tex]:

[tex]y = - (- 3) ^ 2 + 3\\y = -9 + 3 =\\y = -6[/tex]

For [tex]x = 0[/tex]:

[tex]y = - (0) ^ 2 + 3\\y = 0 + 3\\y = 3[/tex]

For [tex]x = 3[/tex]:

[tex]y = - (3) ^ 2 + 3\\y = -9 + 3\\y = -6[/tex]

Answer:

[tex](-3, -6)\\(0,3)\\(3, -6)[/tex]

Answer:

C. increases rapidly.

Step-by-step explanation:

tan(θ) =  sin(θ)/cos(θ)

Now, when sin 90 = 1

and cos 90 = 0

so, tan(90) = 1/0 = not defined.

(1/0 is infinity and its value is not defined)

So, when angle θ increases to 90°, then the value of tan(θ) increases rapidly, as shown in the figure below.

Answer:

  1/3

Step-by-step explanation:

1/2 of 2/3 = 1/2·2/3 = (1·2)/(2·3) = 1/3

Answer:

a) 4/5

b) 1/3

Step-by-step explanation:

Put the numbers in the formula and evaluate.

a) pitch = (rise)/(1/2·span)

= (12 ft)/(1/2·30 ft) = 12/15 = 4/5

___

b) pitch = (4 ft)/(1/2·24 ft) = 4/12 = 1/3

The pitch of a roof is calculated by dividing the rise of the roof by half the span of the roof. The pitch of a roof with a 12 feet rise and 30 feet span is 0.8. The pitch of a roof with a 4 feet rise and 24 feet span is 0.333.

The pitch of a roof can be calculated using the equation: pitch = rise of roof / (½ span of roof).

For the first part of your question, where the rise is 12 feet and the span is 30 feet, you substitute these values into the above equation: pitch = 12 / (½ * 30) = 0.8

For the second part, where the rise is 4 feet and the span is 24 feet, when these values substituted into the equation you get: pitch = 4 / (½ * 24) = 0.333

So, the pitch of the first roof is 0.8 and the pitch of the second roof is 0.333

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Answer:

large box weights 40 pounds & small box weights 35 pounds

Step-by-step explanation:

We can write 2 equations and solve simultaneous.

Let x be weight of large box and y be weight of small box

"The combined weight of a large box and a small box is 75 pounds.":

[tex]x+y=75[/tex]

"The truck is transporting 70  large boxes and 65 small boxes. If the truck is carrying a total of 5075  pounds":

[tex]70x+65y=5075[/tex]

Now we can solve for x in the first equation, which is x = 75 -y

We now substitute this into the 2nd equation and solve for y:

[tex]70x+65y=5075\\70(75-y)+65y=5075\\5250-70y+65y=5075\\-5y=5075-5250\\-5y=-175\\y=35[/tex]

Using y = 35 and plugging that into the 1st equation, we can solve for x:

[tex]x+y=75\\x+35= 75\\x= 75 - 35\\x=40[/tex]

Hence, large box weights 40 pounds & small box weights 35 pounds

1) 29

2) -15

//Hope this helps.

Answer:

1).   (3² - 4²) = 29

2).  5a - 7b + b² = -15

Step-by-step explanation:

Points to remember

1) (a² - b²) = (a + b) (a - b)

Solve 22 - (3² - 4²)

To find (3² - 4²)

(3² - 4²) can be written as,

(3² - 4²) = (3 + 4 ) (3 - 4) = 7 * -1 = -7

22 - (3² - 4²)  = 22 - (-7) = 22 + 7 = 29

22 - (3² - 4²) = 29

To solve 5a - 7b + b²for a = -1 and b= 5

Substitute the value of a and b in given expression we get,

5a - 7b + b² = (5 * -1) - (7 * 5) + 5²

= -5 - 35  + 25 = -15

Therefore 5a - 7b + b² = -15

Answer:

[tex]x=12[/tex] (12 nickels))

[tex]y=5[/tex] (5 dimes)

Step-by-step explanation:

Based on the information given in the problem, you can set up the following system of linear equations, where:

x: number of nickels.

y: number of dimes.

Then

FIRST EQUATION: [tex]x+y=17[/tex]

SECOND EQUATION: [tex]5x+10y=110[/tex]  (in cents)

You can apply the Substitution method:

- Solve for x from the first equation.

- Substitute into the second equation and solve for y.

 Then:

[tex]x=17-y[/tex]

[tex]5(17-y)+10y=110\\85-5y+10y=110\\5y=25\\y=5[/tex]

- Find x:

[tex]x=17-5\\x=12[/tex]

Answer:

Step-by-step explanation:

The problem can be solved by putting the equation into vertex form.

  s - 6 = -16(t^2 -15/4t)

  s -6 -16(15/8)^2 = -16(t^2 -15/4t +(15/8)^2) . . . . add the square of half the x-coefficient inside parentheses and equivalent amount on the other side of the equation

  s -62.25 = -16(t -15/8)^2 . . . . write as a square, simplify the constant

  s = -16(t -1.875)^2 +62.25 . . . . put in vertex form

The vertex of the downward-opening parabola is at (t, s) = (1.875, 62.25).

1. It took 1.875 seconds for the ball to reach its maximum height.

2. The ball went up to 62.25 feet.

Given a markup of 80% and a selling price of $100.80, the cost price is approximately $56. Subtracting this from the selling price yields a cost to store of $44.80.

To find the cost to store, we need to first calculate the cost price using the selling price and the markup percentage.

Given:

- Markup percentage: 80%

- Selling price: $100.80

Step 1: Calculate the cost price using the formula:

[tex]\[ \text{Cost Price} = \frac{\text{Selling Price}}{1 + \frac{\text{Markup Percentage}}{100}} \][/tex]

Substitute the given values:

[tex]\[ \text{Cost Price} = \frac{100.80}{1 + \frac{80}{100}} \][/tex]

[tex]\[ \text{Cost Price} = \frac{100.80}{1 + 0.80} \][/tex]

[tex]\[ \text{Cost Price} = \frac{100.80}{1.80} \][/tex]

[tex]\[ \text{Cost Price} ≈ 56 \][/tex]

Step 2: Calculate the cost to store by subtracting the cost price from the selling price:

[tex]\[ \text{Cost to store} = \text{Selling Price} - \text{Cost Price} \][/tex]

[tex]\[ \text{Cost to store} = 100.80 - 56 \][/tex]

[tex]\[ \text{Cost to store} = 44.80 \][/tex]

So, the cost to store is $44.80.

See the attached picture for the complete answers to both questions:

Question A asked how much more water was required to fill the tank, so you need to calculate the full volume, then subtract 3/4 of the volume.

Question B, volume is cubed, so you need to cube the scale factor of 3. 3^3 = 27, so the larger tank is 27 times more.

Question C) Area is squared so you need to square the scale factor. Surface area would be 9 times more. 3^2 = 9

For the 2nd question, calculate the volume of each box and divide to find the number of boxes.

Then see the picture for total surface area for part B.

Total number of people: 65 + 85 +38 +12 = 200

Total male =  65 +38 = 103

Total women: = 200-103 = 97

Total Type A = 65+85 = 150

Total Type B = 38+12 = 50

P(male or type b ) = 103/200 + 50/200 - 38/200 = 0.515 + 0.25 - 0.19 = 0.765 - 0.19 = 0.575

P(male | type b )  = 38/50 = 0.76

The answer is P(male or type b ) < P(male | type b )

Answer:

  B)  maximum at 15

Step-by-step explanation:

Factor the leading coefficient from the first two terms.

  -(x^2 +6x) +6

Add the square of half the x-coefficient inside parentheses and subtract the same quantity outside.

  -(x^2 +6x +9) +6 -(-9)

  -(x +3)^2 +15

Compared to the form

  a(x -h)^2 +k

we find a=-1, h=-3, k=15. The negative vertical scale factor (a=-1) means the parabola opens downward. The vertex is located at (h, k) = (-3, 15).

The maximum value is 15.

Answer:

B) 5000x^2

Step-by-step explanation:

The exponent of x starts at 5 in the first term and decreases by 1 as the term numbers increase by 1. Hence, the 4th term will have 2 as the exponent of x. (It will have 5-2=3 as the exponent of 5.)

Pascal's triangle tells you the coefficients of the 6 terms in the expansion will be 1, 5, 10, 10, 5, 1, so the 4th term has a coefficient of 10.

The 4th term will be ...

10(2x)^2(5)^3 = 10·4·125·x^2 = 5000x^2

Dividing the total batter of 12 cups by the amount needed per muffin, which is 2/3 cup, reveals that the baker made 18 muffins. The answer is option C.

To find out how many muffins the baker made, we need to use the information given about the amount of batter used and how much batter is required for one muffin. According to the problem, the baker used 12 cups of batter in total and each muffin requires 2/3 cup of batter.

We start by setting up a simple division to find the answer:

(Total amount of batter) \/ (Amount of batter per muffin) = Number of muffins

12 cups \/ 2/3 cups per muffin = 18 muffins

This calculation reveals that the baker made 18 muffins. Therefore, the correct answer is (C) 18 muffins.

Answer:

B. 2x – 1 = 13 and x = 7

Step-by-step explanation:

We are given 4 equations and a solution for each. We have to tell which of the given solution satisfies the given equation.

Option A.

2x -1 = 13 and x = 6

Using this value in the equation, we get:

2(6) -1 = 13

12 - 1 = 13

11 = 13, which is not true. Hence this option is not valid

Option B.

2x - 1 = 13 and x = 7

Using the value in the equation, we get:

2(7) - 1 =13

14 - 1 =13

13 = 13, which is true. Hence this option is valid.

Option C.

2x + 1 =13 and x = 7

Using the value in the equation, we get:

2(7) + 1 = 13

15 = 13, which is not true. So this option is not valid

Option D.

2x - 1 = 13 and x = 11

Using this value in the equation, we get:

2(11) - 1 = 13

21 = 13, which is not true. Hence this option is not valid.

Answer:

s + 10

Step-by-step explanation:

Ruben has s number of shoes. Then, 10 more shoes that Ruben has is

s + 10

Final answer:

To write an expression for 'ten more shoes than Ruben,' define a variable for Ruben's shoes, for example, R, and then add 10. The expression is R + 10.

Explanation:

If you want to write an expression for "ten more shoes than Ruben," you need to start by defining a variable for the number of shoes that Ruben has. Let's say that R represents the number of shoes Ruben has. Then, to express "ten more shoes than Ruben," you simply add 10 to that variable. Therefore, the expression would be R + 10. This straightforward notation clarifies the relationship between Ruben's shoe count and the additional ten pairs. By employing variables and simple arithmetic operations, such as addition, this expression succinctly conveys the concept of having ten more shoes than Ruben, facilitating mathematical communication and problem-solving.

Answer:

7. (4x +10)/(x^3 +3x^2 -16x -48)

9. -320/93

Step-by-step explanation:

7. As with adding any fractions, first you find a common denominator. When the fractions are rational expressions, it often helps to factor the denominators.

6/(x^2 -16) -2/(x^2 -x -12) = 6/((x -4)(x +4)) -2/((x -4)(x +3))

= (6(x +3) -2(x +4))/((x -4)(x +3)(x +4)) . . . . . using a common denominator

= (6x +18 -2x -8)/((x -4)(x +3)(x +4))

= (4x +10)/((x^2 -16)(x +3))

= (4x +10)/(x^3 +3x^2 -16x -48)

_____

9. First you simplify the denominator:

2/25 -5/16 = (2·16 -5·25)/(25·16) = -93/400

Then you perform the division. This can be done by multiplying by the inverse of the denominator.

(4/5)/(2/5 -5/16) = (4/5)·(-400/93) = -320/93

Answer:

  63.7%

Step-by-step explanation:

If a point is chosen from the circle at random, the probability it came from the shaded region is the ratio of the area of the shaded region (square) to the area of the circle.

  area of square = (1/2)(diagonal)^2 = (1/2)·(2 in)^2 = 2 in^2

  area of circle = πr^2 = π·(1 in)^2 = π in^2

Fraction of circle that is shaded = (square area)/(circle area) = 2/π ≈ 63.7%

63.7% good luck hope it’s right

Try suggested solution, note the answers are marked with green colour.

Answer:

Alan: 200 m/min

Brian: 150 m/min

Step-by-step explanation:

Let "a" and "b" represent Alan and Brian's rates in meters per minute, respectively. The rate at which Alan is lapping Brian is the difference in their rates:

a - b = (450 m)/(9 min) = 50 m/min

This lets us write Brian's rate in terms of Alan's as ...

a -50 = b

The time to complete one oval differs by 3/4 minute (45 seconds), so we have ...

time = distance/speed

450/b - 450/a = 3/4

Multiplying by 4ab/3 gives ...

600(a -b) = ab

Substituting from above, we can rewrite this as ...

600·50 = a(a -50)

a^2 -50a -30000 = 0 . . . . . quadratic rearranged to standard form

(a -200)(a +150) = 0 . . . . . . .factored

a = 200 or -150 . . . . only the positive solution is useful here

Alan's rate is 200 m/min; Brian's rate is 150 m/min.

_____

Check

It takes Alan 450/200 min = 2.25 min to complete one oval. It takes Brian 450/150 = 3 min to complete one oval. That is 3-2.25 = 0.75 min = 45 seconds longer than Alan.

After 9 minutes, Alan will have gone (200 m/min)·(9 min) = 1800 m = 4 laps, while Brian will have gone (150 m/min)·(9 min) = 1350 m = 3 laps. Hence Alan will overtake Brian at the 9-minute mark.

Answer:

Part 1) The base is [tex]25\ in[/tex]  and the height is [tex]10\ in[/tex]

Part 2) The base is [tex]7.5\ in[/tex]  and the height is [tex]18.75\ in[/tex]

Step-by-step explanation:

case 1) Right isosceles triangle of the left

Let

x------> the base of the rectangle

y----> the height of the rectangle

Remember that

In a right isosceles triangle the lengths of the legs of the triangle is the same

[tex]y+x+y=45[/tex]

[tex]2y+x=45[/tex] ----> equation A

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

[tex]x=2.5y[/tex] -----> equation B

substitute equation B in the equation A

[tex]2y+2.5y=45[/tex]

[tex]4.5y=45[/tex]

[tex]y=10\ in[/tex]

Find the value of x

[tex]x=2.5(10)=25\ in[/tex]

case 2) Right isosceles triangle of the right

Let

x------> the base of the rectangle

y----> the height of the rectangle

Remember that

In a right isosceles triangle the lengths of the legs of the triangle is the same

[tex]y+x+y=45[/tex]

[tex]2y+x=45[/tex] ----> equation A

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

[tex]y=2.5x[/tex] -----> equation B

substitute equation B in the equation A

[tex]2(2.5x)+x=45[/tex]

[tex]6x=45[/tex]

[tex]x=7.5\ in[/tex]

Find the value of y

[tex]y=2.5(7.5)=18.75\ in[/tex]

In Case 1, the sides of the rectangle are 5x, 45, 45, and 2x. In Case 2, the sides of the rectangle are 5x, 45/sqrt(2), 45/sqrt(2), and 2x.

Let's denote the length of one side of the rectangle as 5x and the length of the other side as 2x. Since the triangle is right isosceles, the two legs are of equal length. Let's denote the length of each leg as a. Using the Pythagorean theorem, we can write an equation: a^2 + a^2 = 45^2. Solving for a, we find that a = 45/sqrt(2).

Case 1: Since two vertices of the rectangle lie on the hypotenuse, the length of the rectangle's side that is parallel to the hypotenuse will be equal to the length of that hypotenuse, which is 45. Therefore, the sides of the rectangle will be 5x, 45, 45, and 2x.

Case 2: Since the two vertices of the rectangle lie on the legs of the triangle, the length of the rectangle's side that is parallel to the hypotenuse will be equal to the lengths of the legs, which is 45/sqrt(2). Therefore, the sides of the rectangle will be 5x, 45/sqrt(2), 45/sqrt(2), and 2x.

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Subtract the two given numbers:

81 - 3 = 78

Divide by the number of missing numbers +1, so divide by 3:

78/3 = 26

This gives you the difference of each number in the sequence, so now subtract 26 from each one:

81 -26 = 55

55 - 26 = 29

29-26 = 3

The two missing numbers are 55 and 29.

To find the missing terms of a geometric sequence, divide each term by a common ratio.

The given sequence is 81, ____, _____, 3.

To find the missing terms, we need to determine the common ratio of the geometric sequence.

Using the given sequence, we can see that each term is obtained by dividing the previous term by a constant factor. Therefore, the common ratio is 81 divided by the first missing term, which is then divided by the second missing term divided by the third missing term, which is then divided by 3.

To find the first missing term, we can divide 81 by the common ratio. Similarly, to find the second missing term, we can divide the first missing term by the common ratio. The missing terms of the geometric sequence are 27 and 9. Therefore, the complete sequence is 81, 27, 9, 3.

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[tex]\rm \dfrac{HL}{JL}=\dfrac{IL}{KL}=\dfrac{HI}{JK}[/tex]

Step-by-step explanation:

Given :

[tex]\rm \bigtriangleup HLI \sim \bigtriangleup JLK[/tex]  by the SSS similarity theorem.

[tex]\rm \dfrac {HL}{JL} = \dfrac{IL}{KL}[/tex]

Solution :

According to SSS postulate,

KL = IL ----- (1)

HL = JL ----- (2)

HI = JK ------ (3)

From equation (1), (2) and (3) we get,

[tex]\rm \dfrac{HL}{JL}=\dfrac{IL}{KL}=\dfrac{HI}{JK}[/tex]

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Triangle HLI has sides identical in measure to the three sides of triangle JLK, then the two triangles are congruent by SSS postulate. Therefore,

[tex]\rm \dfrac{HL}{JK}=\dfrac{IL}{KL}=\dfrac{HI}{JK}[/tex]

Given :

[tex]\rm \Delta HLI \sim \Delta JLK[/tex] by the SSS similarity.

According to SSS (Side Side Side) postulate:

If all three sides of a triangle are congruent to corresponding three sides of other triangle then the two triangles are congruent.

If triangle HLI has sides identical in measure to the three sides of triangle JLK, then the two triangles are congruent by SSS postulate.

Therefore, from triangle HLI and triangle JLK we get,

KL = IL  ----- (1)

HL = JL ----- (2)

HI = JK ------ (3)

From equation (1) , (2) and (3) we get,

[tex]\rm \dfrac{HL}{JK}=\dfrac{IL}{KL}=\dfrac{HI}{JK}[/tex]

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Answer:

  9x² -6xy +y²

Step-by-step explanation:

The square of the binomial can be written directly using the relationship ...

  (a +b)² = a² +2ab +b²

where you have a=3x and b=-y.

  area = (3x -y)² = (3x)² + 2(3x)(-y) + (-y)²

  area = 9x² -6xy +y²

Here is your answer

D. [tex]\frac{8788}{3}[/tex]π [tex]{units}^{3}[/tex]

REASON:

Formula for volume of sphere

= 4/3×π× r^3

Here, r= 13

So,

volume=4/3×π× 13^3

= 4/3×π× 2197

= 8788/3 ×π

HOPE IT IS USEFUL

The volume of the sphere is [tex]\bold{\frac{8788}{3} \pi ~~unit^3}[/tex]

The correct answer is option (D).

V = (4/3) × π × r³, where 'r' is the radius of the sphere

For given example,

radius of the sphere (R) = 13

Using the formula o the volume of the sphere,

V = (4/3) × π × R³

V = (4/3) × π × 13³

V = 8788/3 π cubic units.

Therefore, the volume of the sphere is [tex]\frac{8788}{3} \pi[/tex] cubic units.

The correct answer is option (D) [tex]\frac{8788}{3} \pi ~~unit^3[/tex]

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Answer:

[tex]A=163.9in^2[/tex]

Step-by-step explanation:

Let's use the regular polygon area formula when given the radius.

[tex]A=\frac{(r^{2})(n)sin(\frac{360}{n})}{2}[/tex]

r is radius

n is the number of sides

Let's plug in the proper values into this equation and simplify:

[tex]A=\frac{(7.74^{2})(7)sin(\frac{360}{7})}{2}[/tex]

[tex]A=\frac{(59.9076)(7)(0.78183148)}{2}[/tex]

[tex]A=\frac{327.86353299}{2}[/tex]

[tex]A=163.93176649[/tex]

Answer:

The correct answer is c.

Step-by-step explanation:

This is because it is the first equation being multiplied by 3 and the second being multiplied by 2.

None of the other examples are multiples of the originals.

Answer:

[tex]10\sqrt{2}[/tex]

Step-by-step explanation:

Pythagorean Theorem

QS^2+SR^2=QR^2

10^2+10^2=QR^2

100+100=QR^2

200=QR^2

[tex]\sqrt{200}=QR[/tex]

[tex]10\sqrt{2}[/tex]

Answer:

In right Δ QRS, right angled at , S.

By Pythagorean theorem

→(Hypotenuse)²=(Altitude)^2 +(Base)²

→ Q S² + SR² = QR²

→10² +10²= QR²

→ QR²=100+100

→ QR²=200

[tex]QR=\sqrt{200}\\\\ QR=\sqrt{10 \times 10 \times 2}\\\\ QR=10\sqrt{2}[/tex]

Option D

Hypotenuse[tex]=10\sqrt{2}[/tex] units