Logan can paint a room in 12 hours. Marcus can paint the same room in 16 hours. How long does it take for both Logan and Marcus to paint the room if they are working together?

The lengths of segments in the circle are determined by the tangent-secant theorem. Given RT = 9 inches and ST = 4 inches, we can use this theorem to find that TU will be approximately 8.06 inches.

In this complex geometry problem involving circles and tangents, there are several relationships at play. You're looking at a circle that has a secant, RT, and a tangent, TU, creating several segments within the figure.

(a) According to the tangent-secant theorem, the squared length of the tangent segment is equal to the difference in the squared lengths of the entire secant and its external part. This can be expressed mathematically as TU² = RT² - ST².

(b) We can use this theorem with the given lengths for RT and ST to find the length of TU. Plugging in, we get TU² = 9² - 4² = 65. The length of TU is the square root of this, which is approximately 8.06 inches.

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The transformation of [tex]y = -\frac12(5)^{x-3[/tex] to [tex]y = \frac12(5)^{x[/tex] is reflection across the x-axis followed by a shift to the left by 3 units

Describing the transformation of f(x) to g(x).

From the question, we have the following parameters that can be used in our computation:

[tex]y = -\frac12(5)^{x-3[/tex]

[tex]y = \frac12(5)^{x[/tex]

Rewrite them as

[tex]f(x) = -\frac12(5)^{x-3[/tex]

[tex]g(x) = \frac12(5)^{x[/tex]

From the above, we have that

f(x) is first reflected across the x-axis

This is represented by the loss of negative factor in g(x)

Then f(x) is shifted left by 3 units

This is represented by the loss of -3 in g(x)

Hence, the transformation of f(x) to g(x) is reflection across the x-axis followed by a shift to the left by 3 units

Given is the expression, x² + 12x + n

It says to find value of 'n' that converts it into a Perfect Square Trinomial.

A perfect square trinomial is a quadratic expression which has a single root with multiplicity 2. i.e. it is of the form :- x² + 2bx + b² = (x + b)²

On comparing it with the given expression, we get :-

So 2b = 12 and n = b²

b = 6

and so b² = 6² = 36.

⇒ n = 36

Hence, x² + 12x + 36 = (x + 6)² i.e. n = 36 is the final answer.

D) T(x,y) = (x, y - 2); The transformation slides the original image a given distance in a given direction. Is the CORRECT answer

Answer:

$18.

Step-by-step explanation:

We have been given that Joe borrowed $900 from Sam for six months. The interest rate charged by Sam is 4%.

We will use Simple interest formula to solve our given problem.

[tex]I=PrT[/tex], where,

[tex]I=\text{Simple interest}[/tex],

[tex]P=\text{Principal amount}[/tex],

[tex]r=\text{Interest rate in decimal form}[/tex],

[tex]T=\text{Time in years}[/tex].

Let us convert our given rate in decimal form.

[tex]4\%=\frac{4}{100}=0.04[/tex]

12 months = 1 year.

1 month = 1/12 year.

6 months = 6/12 year = 1/2 year = 0.5 year.

Upon substituting our given values in simple interest formula we will get,

[tex]I=900*0.04*0.5[/tex]

[tex]I=900*0.02[/tex]

[tex]I=18[/tex]

Therefore, Sam will earn $18 if he charge Joe a simple interest rate of 4 percent.

The answer will be 27/35

Probability is the number of outcomes out of total outcomes present or  the extent to which something is likely to happen. In the given problem there are two dices which means there are total 36 outcomes because each dice has 6 sides. So the probability of the given case will be

Probability of getting two doubles on both dice = 1/36

Probability of getting just one double = 5/36

Similarly

Probability of getting no doubles on both dice is = 5/6.  

So in order to find the probability we add the products of the probabilities and solve:

12/36 + 5*(3-x)/36 - 5*x/6 = 0

x = 27/35

Which simply means the player should lose 27/35 points for not rolling a double.

This the answer hope it helps!!

Answer:

[All reals]............

The volume of one paper cone if the diameter is 4 inches and the height, is 7 inches is equal to 29in³.

We have given that,

Mrs. McDonnell is making 25 paper cones to fill with popcorn for her daughter's birthday party.

The volume of one cone would be 29 in³.

The formula for the volume of a cone is V=(1/3)πr²h.  

Substituting our information we have:

V=(1/3)(3.14)(4/2)²(7)

We divide the diameter by 2 because the radius is half of the diameter.  This gives us:

V=1/3(3.14)(28)

V= 29.306

V≈ 29in³

Therefore the volume of one paper cone if the diameter is 4 inches and the height, is 7 inches is equal to 29in³.

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Answer:

The solution set is [tex]x=-9[/tex]

Step-by-step explanation:

we have

[tex](x+9)(x+9)=0[/tex]

we know that

[tex](x+9)(x+9)=x^{2}+18x+81[/tex]

so

[tex]x^{2}+18x+81=0[/tex]

The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to

[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]

in this problem we have

[tex]x^{2}+18x+81=0[/tex]

so

[tex]a=1\\b=18\\c=81[/tex]

substitute in the formula

[tex]x=\frac{-18(+/-)\sqrt{18^{2}-4(1)(81)}} {2(1)}[/tex]

[tex]x=\frac{-18(+/-)\sqrt{0}} {2}[/tex]

The radicand is equal to zero, therefore, the solution has only one real solution

[tex]x=\frac{-18} {2}=-9[/tex]

Answer:

Option A. [tex]4[/tex]

Step-by-step explanation:

we know that

The sum of a geometric series is equal to

[tex]Sum=a(\frac{1-r^{n}}{1-r})[/tex]

where

a is the first term

r is the common ratio

n is the number of terms

In this problem we have

[tex]a=2,r=0.5,n=16[/tex]

substitute the values

[tex]Sum=2(\frac{1-0.5^{16}}{1-0.5})=4[/tex]

Final answer:

To find the area of the rectangle, first solve for the width using the perimeter formula. The width is found to be 7 ft. Then calculate the length as w + 2, which is 9 ft. The area is the product of length and width, resulting in 63 square feet.

Explanation:

To solve this problem, we need to set up equations based on the information given about the rectangle's dimensions and the perimeter.

Step 1: Define Variables

Let w be the width of the rectangle in feet. Consequently, the length l will be w + 2 feet, since the length is 2 feet longer than the width.

Step 2: Perimeter Equation

The perimeter (P) of a rectangle is given by 2(l + w). We can use the perimeter given (32 ft) to create an equation: 2(w + w + 2) = 32.

Step 3: Solve for w

Combine like terms and divide both sides by 2 to find w:
4w + 4 = 32
w = 7 ft.

Step 4: Find l

The length is w + 2, which is 7 ft + 2 ft = 9 ft.

Step 5: Calculate Area

The area of a rectangle is l × w. Therefore, the area is 9 ft × 7 ft = 63 square feet.

The number of sundaes they can serve with 7 cups of sprinkles is 56.

The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.

Given that, Matt's Ice Cream Shoppe has 7 cups of sprinkles to use on sundaes for the rest of the day.

Number of sundaes = Total quantity of sprinkles/Number of cups of sprinkles in each serving

= 7÷ 1/8

= 7×8

= 56

Therefore, the number of sundaes they can serve with 7 cups of sprinkles is 56.

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The value of y is 3

Solving linear equation mean calculating the unknown variable from the equation.

Let the linear equation : y = mx + c

If we draw the above equation on Cartesian Coordinates , it will be a straight line with :

m → gradient of the line

( 0 , c ) → y - intercept

Gradient of the line could also be calculated from two arbitrary points on line ( x₁ , y₁ ) and ( x₂ , y₂ ) with the formula :

[tex]\large {\boxed {m = \frac{y_2 - y_1}{x_2 - x_1}} }[/tex]

If point ( x₁ , y₁ ) is on the line with gradient m , the equation of the line will be :

[tex]\large {\boxed {y - y_1 = m ( x - x_1 )} }[/tex]

Let us tackle the problem.

Firstly , we will calculate the gradient of the line that passes through the point S( -5 , 0 ) and T( 5 , 2 ) .

[tex]m_{ST} = \frac{y_2 - y_1}{x_2 - x_1}[/tex]

[tex]m_{ST} = \frac{2 - 0}{5 - (-5)}[/tex]

[tex]m_{ST} = \frac{2}{10}[/tex]

[tex]\large {\boxed {m_{ST} = \frac{1}{5} } }[/tex]

Next , we will calculate the gradient of the line that passes through the point V( 0 , -2 ) and W( -1 , y ) .

[tex]m_{VW} = \frac{y_2 - y_1}{x_2 - x_1}[/tex]

[tex]m_{VW} = \frac{y - (-2)}{-1 - 0}[/tex]

[tex]\large {\boxed {m_{VW} = \frac{y + 2}{-1} } }[/tex]

The conditions of the line perpendicular to each other will satisfy the following formula.

[tex]m_{ST} \times m_{VW} = -1[/tex]

[tex]\frac{1}{5} \times \frac{y + 2}{-1} = -1[/tex]

[tex]\frac{y + 2}{-5} = -1[/tex]

[tex]y + 2 = -5 \times -1[/tex]

[tex]y + 2 = 5[/tex]

[tex]y = 5 - 2[/tex]

[tex]\large {\boxed {y = 3} }[/tex]

Grade: High School

Subject: Mathematics

Chapter: Linear Equations

Keywords: Linear , Equations , 1 , Variable , Line , Gradient , Point