Use x=1+r , where r is the interest rate paid each year. Write a model polynomial, C(x) . Represent the final amount of a 7-year investment if $5,000 was deposited at the beginning of the first year, and $2,000 was deposited at the beginning of the third year.(1 point)

Question

Responses

C(x)=5,000x7+2,000x4
upper C left parenthesis x right parenthesis equals 5,000 x superscript 7 baseline plus 2,000 x superscript 4 baseline

C(x)=5,000x+2,000x3
upper C left parenthesis x right parenthesis equals 5,000 x plus 2,000 x cubed

C(x)=5,000x7+2,000x3
upper C left parenthesis x right parenthesis equals 5,000 x superscript 7 baseline plus 2,000 x cubed

C(x)=5,000x7+7,000x4

Answer 1

To represent the final amount of the investment as a polynomial, we need to consider the contributions of both deposits over the 7 years.

1. The first deposit of $5,000 is made at the beginning of the first year, so it will earn interest for 7 years. Its contribution can be represented as \( 5,000 \cdot x^7 \).

2. The second deposit of $2,000 is made at the beginning of the third year, so it will earn interest for 5 years (from the start of year 3 to the end of year 7). Its contribution can be represented as \( 2,000 \cdot x^5 \).

Thus, the model polynomial \( C(x) \) would be: \[ C(x) = 5,000x^7 + 2,000x^5 \]

None of the options you provided match this expression exactly, but based on the context of what we derived, the correct representation of the polynomial form considering how the investments grow over the years would be similar to:

Answer: \( C(x) = 5,000x^7 + 2,000x^5 \).

Please verify if any of the provided options might be misprinted. If we must select the closest option, then "C(x)=5,000x^7+2,000x^4" might be the closest but is not accurate because the power for the second term should be 5, not 4.

Answer 2

The equation A(w)=w2+5w represents the relationship between the area in square units

and the width of a rectangle whose length is 5 units longer than its width.

Select the sentence that describes an accurate relationship between A
and w
.

(1 point)
Responses

A(w)
increases as w
increases when w>0
.
cap A times w increases as w increases when w is greater than 0.

A(w)
has a maximum value at w=5
.
cap A times w has a maximum value at w is equal to 5.

A(w)
increases as w
increases for 0<w<5
.
cap A times w increases as w increases for 0<w<5
.

A(w)
has a minimum value at w=5
.

Answer 3

To analyze the equation \( A(w) = w^2 + 5w \), we can determine its behavior and relationships:

1. Type of Function: The equation represents a quadratic function in the standard form \( A(w) = aw^2 + bw + c \), where \( a = 1 \), \( b = 5 \), and \( c = 0 \).

2. Shape of the Parabola: Since \( a > 0 \), the parabola opens upwards.

3. Finding the Vertex: The vertex of a parabola given by \( A(w) = aw^2 + bw + c \) can be found using the formula: \[ w = -\frac{b}{2a} \] In this case: \[ w = -\frac{5}{2 \cdot 1} = -\frac{5}{2} \] which is negative. This means that the function \( A(w) \) does not have a maximum value since it opens upwards, and it will keep increasing as \( w \) increases.

4. Behavior for Positive Width: Since we are interested in \( w > 0 \), we need to check how \( A(w) \) behaves in that interval. Since the vertex occurs at \( w = -\frac{5}{2} \), and for \( w > 0 \), \( A(w) \) will continuously increase.

Analyzing the Responses:

• A(w) increases as w increases when w > 0: This statement is correct.

• A(w) has a maximum value at w = 5: This statement is incorrect because the function has no maximum as it increases indefinitely.

• A(w) increases as w increases for 0 < w < 5: This statement is also true but does not encompass the entire range beyond \( w = 5 \). A(w) increases for all \( w > 0\).

• A(w) has a minimum value at w = 5: This is incorrect; since the function is quadratic and opens upwards, it doesn't reach a minimum at that point (the minimum occurs at \( w < 0 \)).

Conclusion:

The best sentence to describe the accurate relationship between \( A \) and \( w \) is: A(w) increases as w increases when w > 0.

Answer 4

The equation V=43πr3 represents the relationship between the volume of a sphere and its radius. What does the end behavior tell you about the relationship between the volume of the sphere and its radius?(1 point)
Responses

As the radius increases to infinity, the volume of the sphere will increase to infinity.
As the radius increases to infinity, the volume of the sphere will increase to infinity.

As the radius increases to infinity, the volume of the sphere will decrease to negative infinity.
As the radius increases to infinity, the volume of the sphere will decrease to negative infinity.

As the radius decreases to negative infinity, the volume of the sphere will increase to infinity.
As the radius decreases to negative infinity, the volume of the sphere will increase to infinity.

As the radius decreases to negative infinity, the volume of the sphere will remain constant.

Answer 5

To analyze the equation \( V = \frac{4}{3} \pi r^3 \), we will explore its end behavior as the radius \( r \) changes.

End Behavior of the Function:

1. Behavior as \( r \) Increases:

   • As \( r \) increases towards positive infinity (\( r \to \infty \)), the term \( r^3 \) also increases towards infinity. Therefore, since \( V \) is directly proportional to \( r^3 \), \( V \) will also increase towards positive infinity.
   • Thus, we can conclude that when the radius grows larger, the volume of the sphere increases without bound.

2. Behavior as \( r \) Decreases:

   • The radius of a sphere cannot be negative, as a negative radius would not have any physical meaning in the context of a sphere. So, we typically consider \( r \geq 0 \).
   • If we consider the behavior for decreasing values of \( r \) (specifically as \( r \) approaches zero), the volume \( V \) approaches 0 (\( V \to 0 \) as \( r \to 0 \)). However, if we think about negative values of \( r \), the equation \( V = \frac{4}{3} \pi r^3 \) would give a negative volume. But again, in physical terms, volume cannot be negative.

Options Analysis:

1. As the radius increases to infinity, the volume of the sphere will increase to infinity. - This statement is true and accurately reflects the behavior of the function.

2. As the radius increases to infinity, the volume of the sphere will decrease to negative infinity. - This statement is false, as we've determined the volume increases, not decreases.

3. As the radius decreases to negative infinity, the volume of the sphere will increase to infinity. - This statement is misleading as, conceptually, the radius cannot decrease to negative infinity in a geometric context.

4. As the radius decreases to negative infinity, the volume of the sphere will remain constant. - This statement is also false, as volume cannot be constant (or meaningful) with negative radii.

Conclusion:

The most accurate statement regarding the volume of the sphere as the radius increases is:

As the radius increases to infinity, the volume of the sphere will increase to infinity.

Answer 6

Brayton wants to invest his high school earnings for the next 7 years. He deposits $1,500 into an investment account at the beginning of the first year, $2,000 at the beginning of the second year, $1,870 at the beginning of the third year, and $2,230 at the beginning of the fourth year. Use x=1+r , where r is the interest rate, and the equation C(x)=1500x7+2000x6+1870x5+2230x4 . The equation represents the relationship between C(x), the value of the investment after 7 years, and its annual interest rate, r. Find the value of the investment account if the interest rate is 2.85%.(1 point)
Responses

$7,716.96
$7,716.96

$7,600.00
$7,600.00

$8,840.80
$8,840.80

$30,314.09

Answer 7

To find the value of Brayton's investment account after 7 years at an interest rate of 2.85%, we'll use the equation:

\[ C(x) = 1500x^7 + 2000x^6 + 1870x^5 + 2230x^4 \]

Where \( x = 1 + r \). Given that \( r = 0.0285 \) (which represents the 2.85% interest rate), we have:

\[ x = 1 + 0.0285 = 1.0285 \]

Now we will calculate \( C(x) \) by substituting \( x = 1.0285 \) into the equation:

1. Calculate \( x^4, x^5, x^6, \) and \( x^7 \):
   • \( x^4 = (1.0285)^4 \)
   • \( x^5 = (1.0285)^5 \)
   • \( x^6 = (1.0285)^6 \)
   • \( x^7 = (1.0285)^7 \)

Using a calculator:

\[ x^4 \approx 1.1173 \] \[ x^5 \approx 1.1492 \] \[ x^6 \approx 1.1820 \] \[ x^7 \approx 1.2156 \]

2. Now we can substitute these values back into \( C(x) \):

\[ C(x) = 1500(1.2156) + 2000(1.1820) + 1870(1.1492) + 2230(1.1173) \]

Calculating each term:

• \( 1500 \times 1.2156 \approx 1823.40 \)
• \( 2000 \times 1.1820 \approx 2364.00 \)
• \( 1870 \times 1.1492 \approx 2145.64 \)
• \( 2230 \times 1.1173 \approx 2492.56 \)

3. Adding these values together:

\[ C(x) \approx 1823.40 + 2364.00 + 2145.64 + 2492.56 \]

Calculating the total:

\[ C(x) \approx 1823.40 + 2364.00 = 4187.40 \] \[ 4187.40 + 2145.64 \approx 6333.04 \] \[ 6333.04 + 2492.56 \approx 8825.60 \]

Final Calculation:

Thus, after calculating, we find:

\[ C(x) \approx 8825.60 \]

This value \($8,825.60\) does not match any of the provided answers directly, but for accuracy, let’s ensure the addition is precise, because it appears close to one of the options.

After performing these calculations, it appears your investment account value at an interest rate of 2.85% after 7 years would be approximately:

Final Answer: The closest option based on my detailed calculations is $8,840.80.