While Joey was standing outside on a sunny day, his shadow was 5 feet long. The shadow of a nearby Oak tree was 12 feet long. If Joey is 6 feet tall, how tall is the Oak tree?

A. 15.6 feet

B. 14.4 feet

C. 10 feet

D. 13.4 feet

Answer:

The domain of f(x) is all real numbers greater than or equal to -2

Step-by-step explanation:

We have been given that

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

Domain is the set of x values for which the function is defined.

The given function is the combination of a square and square root function.

Square function is defined for all real values. Whereas, the square root function is defined for only positive values.

Therefore, the function is defined when

[tex]x + 2\geq0[/tex]

Subtract 2 to both sides

[tex]x\geq-2[/tex]

Hence, the domain of f(x) is all real numbers greater than or equal to -2

Martina would need $1,214,720 to maintain her current lifestyle for 20 years post-retirement, based on a gross weekly pay of $1,168. None of the provided options match this calculated amount, suggesting a need for a reassessment.

To determine how much Martina should save by the time she retires, we must first understand her annual consumption.

If Martina's gross weekly pay is $1168, her annual income is calculated as follows:

Annual Income = 52 weeks/year × $1168/week = $60,736/year.

Assuming Martina plans to spend the entire amount to maintain her current lifestyle during retirement, and assuming she lives 20 years post-retirement, the total amount she would need is:

Total Retirement Savings Needed = $60,736/year × 20 years = $1,214,720.

Given the provided options:

None of the provided choices match the calculated amount of $1,214,720 needed for maintaining her current lifestyle over 20 years of retirement. Therefore, additional context is needed, or a reassessment of Martina's retirement duration and lifestyle costs should be considered.

Answer:

[tex](fg)(x)=4x^3+12x^2-5x-15[/tex]

First option is correct.

Step-by-step explanation:

We have been given that

[tex]f(x)=4x^2-5[/tex]

[tex]g(x)=x+3[/tex]

We know that [tex](fg)(x)=f(x)g(x)[/tex]

Hence, we have to find the product of f(x) and g(x)

[tex](fg)(x)=(4x^2-5)(x+3)[/tex]

Apply distributive property

[tex](fg)(x)=4x^2\cdot x+4x^2\cdot3-5x-5\cdot3[/tex]

Simplifying, we get

[tex](fg)(x)=4x^3+12x^2-5x-15[/tex]

First option is correct.

The number of roots a polynomial will have can be determined by its degree, assuming that we are considering all possible roots including complex roots and counting each root according to its multiplicity.

For example, a linear polynomial (degree 1) such as [tex]\( ax + b = 0 \)[/tex] will have exactly one root. A quadratic polynomial (degree 2) like [tex]\( ax^2 + bx + c = 0 \)[/tex] will have two roots, which could be real or complex. Similarly, a cubic polynomial (degree 3) will have three roots, and so on.

If the coefficients of the polynomial are real numbers, then any complex roots will occur in conjugate pairs. This means that a real polynomial of odd degree will always have at least one real root.

To summarize, the number of roots of a polynomial equation is equal to the degree of the polynomial, provided that:

1. We include all real and complex roots.

2. We count each root according to its multiplicity (the number of times it is repeated).

3. The polynomial is non-constant (degree greater than 0).

Dad's age after his birthday is not a prime number because before his birthday his age was odd, and adding 1 to it made it even. All even numbers greater than 2 are not prime as they are divisible by 2. The new age's prime factors summed with 1 is also divisible by 3, confirming its non-primality.

Before Dad's birthday, dividing his age by 2 left a remainder of 1, which means his age was an odd number. When an odd number has 1 added to it (as occurs on a birthday), it becomes an even number. Since all even numbers except 2 are not prime because they are divisible by 2, we can conclude that Dad's new age cannot be a prime number.

Prime numbers are those that are only divisible by 1 and themselves, and the smallest even prime is 2. Therefore, if Dad's age increased by 1 and became an even number greater than 2, it cannot be prime due to the divisibility rule.

Considering the character of ages having the sum of their prime factors (including one) divisible by 3, and using the modulus operator, we can affirm that the age after the birthday (which is even) conforms to this pattern, hence, verifying its non-primality.

Final answer:

To find out how much vinegar Bob needs for 2 3/4 recipes, simply multiply 2/5 cup (the amount used for one recipe) by 2 3/4. The answer is 1.1 cups of vinegar.

Explanation:

To calculate the amount of vinegar Bob would use to make 2 3/4 of his salad dressing recipes, we first need to determine how much vinegar is used in one full recipe. We then multiply that amount by 2 3/4 to find the total vinegar needed for the larger batch.

Bob uses 2/5 of a cup of vinegar for one recipe. To make 2 3/4 recipes, we perform the following calculation:

Therefore, Bob would use 1.1 cups of vinegar to make 2 3/4 of his salad dressing recipes.

The length and width of the given rectangle with area of 182 square units is; Length = 26 units and width = 7 units

Let the length of the rectangle be x

Let the width of rectangle be y

Let the area of rectangle be A

Formula for area of rectangle is;

A = length * width

Thus;

A = xy

We are told that area is 182 square units. Thus;

xy = 182 sq.units   -----(eq 1)

Also the length of the rectangle is 5 more than 3 times its width. Thus;

x = 3y + 5   ------(eq 2)

Put 3y + 5 for x in eq 1 to get;

y(5 + 3y) = 182

3y² + 5y = 182

3y² + 5y - 182 = 0

Using online quadratic equation calculator gives; y = 7

Thus;  x = 182/7

x = 26

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The sand will be 2 inches deep in the new terrarium if Zach keeps his pet chameleon Pinky in a terrarium with the dimensions of 8 x 20. there are three inches of sand at the bottom of the terrarium.

It is defined as the six-faced shape, a type of hexahedron in geometry.

It is a three-dimensional shape.

We have:

Old terrarium dimensions = 8 x 20 x 3

As the sand height is 3 inches

Volume of the old terrarium = 8 x 20 x 3 ⇒ 480 cubic inches

Let's suppose the height of the sand in the new terrarium is h inches

So;

The volume of the new terrarium = 10 x 24 xh ⇒ 240h cubic inches

Since the volume remains same;

480 = 240h

h = 2 inches

Thus, the sand will be 2 inches deep in the new terrarium if Zach keeps his pet chameleon Pinky in a terrarium with the dimensions of 8 x 20. there are three inches of sand at the bottom of the terrarium.

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Answer:

Volume of the sphere(V) is given by:

[tex]V = \frac{4}{3} \pi r^3[/tex]       ....[1]

where,

r is the radius of the sphere.

As per the statement:

The diameter of a sphere is 4 centimeters.

Formula for the diameter(d) is given by:

[tex]d = 2r[/tex]

⇒[tex]4 = 2r[/tex]

Divide both sides by 2 we have;

2 = r

or

r = 2 cm

Substitute in [1]  and use 3.14 for pi. we have;

[tex]V = \frac{4}{3} \cdot 3.14 \cdot 2^3 = \frac{4}{3} \cdot 3.14 \cdot 8[/tex]

Simplify:

V = 33.4933333 cubic cm

therefore,  the volume of the sphere to the nearest tenths is, [tex]33.5 cm^3[/tex]

To maximize the apple yield from an acre, the optimal number of apple trees to plant is 24 trees.

In this problem, we need to maximize the yield of apples from an acre of an apple farm. The yield of apples from an acre is given by the product of the number of trees and the number of bushels per tree. The number of bushels of apples per tree decreases by 1 bushel every time an additional tree is added due to crowding. In mathematical terms, this can be expressed as the equation Y = N * (31 - (N - 17)), where Y is the total yield, and N is the number of trees per acre.

To find the maximum yield, we differentiate this equation with respect to N and set the derivative equal to zero. This gives us N = 24. Therefore, the optimal number of trees to plant per acre in order to maximize apple yield is 24 trees.

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To maximize the yield of apples per acre, plant 24 trees. This balances the number of trees and the yield per tree due to crowding.

To find the number of trees that should be planted per acre to get the highest yield, we need to model the yield per acre as a function of the number of trees planted and then find the maximum value of this function.

1. Define the variables:

  - Let ( n ) be the number of trees planted per acre.

  - The yield per tree decreases by 1 bushel for each additional tree planted, starting from 31 bushels per tree when there are 17 trees planted.

2. Model the yield per tree:

  - If ( n = 17 ), the yield per tree is 31 bushels.

  - For each additional tree, the yield decreases by 1 bushel, so the yield per tree can be expressed as ( 31 - (n - 17) ).

  Simplifying this, we get:

   Yield per tree = 31 - n + 17 = 48 - n

3. Model the total yield per acre:

  - The total yield ( Y ) per acre is the number of trees ( n ) times the yield per tree.

 Y = n × (48 - n)

4. Formulate the quadratic function:

  Y = 48n - [tex]n^2[/tex]

5. Find the maximum yield:

  - This is a quadratic function of the form [tex]\( Y = -n^2 + 48n \)[/tex], which is a downward-opening parabola. The maximum value of \( Y \) occurs at the vertex of the parabola.

  - The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] occurs at [tex]\( x = -\frac{b}{2a} \)[/tex] . In our case, ( a = -1 ) and ( b = 48 ).

 [tex]\[ n = -\frac{48}{2(-1)} = \frac{48}{2} = 24 \][/tex]

So, to get the highest yield, 24 trees should be planted per acre.

Answer:

Step-by-step explanation:

Given: The measure of the arc QU=88°, ∠QUA=111°.

To find: The measure of the arc AU.

Construction: Join QX and UX and AX, where X is the center of the circle such that ∠UXA=x.

Solution: It is given that  The measure of the arc QU=88°, ∠QUA=111°.

Now, we know that The measure of a minor arc is the same as the measure of the central angle that corresponds to it, therefore

If the minor arc QU=88°, then ∠QXU=88°.

Also, we know that the major central angle is double of the inscribed angle, thus

∠QXA=2(∠QUA)

⇒∠QXA=2(111°)

⇒∠QXA=222°

Using the angle sum property for a point X, we have

∠QXA+∠QXA=360°

⇒222°+88+x=360°

⇒310°+x=360°

⇒x=50°.

Again, The measure of a minor arc is the same as the measure of the central angle that corresponds to it, thus

The measure of the arc AU is same as ∠UXA that is 50°,therefore, the measure of AU is 50°.

Hence, option B is correct.

Answer:

tan x 7/24

Step-by-step explanation:

I took the test this is the answer

The answer is:

y = -x + 13

Work/explanation:

First, we will write the equation in the form of [tex]\sf{y-y_1=m(x-x_1)}[/tex], which is point slope.

Plug in the data

[tex]\large\begin{gathered}\sf{y-15=-1(x-(-2)}\\\sf{y-15=-1(x+2)}\\\sf{y-15=-1x-2}\\\sf{y-15=-x-2}\\\sf{y=-x-2+15}\\\sf{y=-x+13}\end{gathered}[/tex]

Hence, the slope intercept is y = -x + 13.

The height of the tree is 7.35 feet.

To find the height of the tree, we need to use similar triangles. Here’s the detailed process:

Understand the problem and convert units: Tanya is 5 feet, 3 inches tall. Convert her height to feet.

  - 3 inches is 0.25 feet.

  - So, Tanya’s height is [tex]\( 5 + 0.25 = 5.25 \)[/tex] feet.

According to diagram

  - A tree T with height h.

  - Tanya is standing at a point B, with her shadow extending to point E.

  - The tree’s shadow extends to point C.

  - The tip of Tanya’s shadow coincides with the tip of the tree’s shadow at point C.

  - Let E be the point where Tanya’s shadow ends.

  - Let EC = x.

  - [tex]\( BE = 2.5 \times EC = 2.5x \)[/tex].

Using similar triangles:

 [tex]\[ \frac{\text{Tanya's height}}{\text{Length of Tanya's shadow}} = \frac{\text{Tree's height}}{\text{Length of tree's shadow}} \][/tex]

[tex]\[ \frac{5.25}{2.5x} = \frac{h}{3.5x} \][/tex]

  Simplify the ratio:

[tex]\[ \frac{5.25}{2.5} = \frac{h}{3.5} \][/tex]

Cross-multiply and solve for h:

[tex]\[ 5.25 \times 3.5 = 2.5 \times h \][/tex]

  Calculate [tex]\( 5.25 \times 3.5 \):[/tex]

 [tex]\[ 5.25 \times 3.5 = 18.375 \][/tex]

  Now, divide by 2.5 to find h:

  [tex]\[ h = \frac{18.375}{2.5} = 7.35 \][/tex]

The complete question is

Tanya wants to find the height of the tree. She walks away from the base of the tree so the tip of her shadow coincides with the tip of the tree's shadow at point C. BE is two and a half times EC. Tanya is 5 feet, 3 inches tall. How tall is the tree in feet?