PLEASE I NEED THIS ASAPPP!!!

For what values of θ on the polar curve r=θ, with 0≤θ≤2π , are the tangent lines horizontal? Vertical?

Bond [772]

Given that $r=\theta$ , then $r'=1$

The slope of a tangent line in the polar coordinate is given by:

$m= \frac{r'\sin\theta+r\cos\theta}{r'\cos\theta-r\sin\theta}$

Thus, we have:

$m= \frac{\sin\theta+\theta\cos\theta}{\cos\theta-\theta\sin\theta}$

Part A:

For horizontal tangent lines, m = 0.

Thus, we have:

$\sin\theta+\theta\cos\theta=0 \\ \\ \theta\cos\theta=-\sin\theta \\ \\ \theta=- \frac{\sin\theta}{\cos\theta} =-\tan\theta$

Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are horizontal are:

θ = 0

θ = 2.02875783811043

θ = 4.91318043943488

Part B:

For vertical tangent lines, $\frac{1}{m} =0$

Thus, we have:

$\cos\theta-\theta\sin\theta=0 \\ \\ \Rightarrow\theta\sin\theta=\cos\theta \\ \\ \Rightarrow\theta= \frac{\cos\theta}{\sin\theta} =\sec\theta$

Therefore, the values of θ on the polar curve r = θ, with 0 ≤ θ ≤ 2π, such that the tangent lines are vertical are:

θ = 4.91718592528713

3 0

3 years ago

Kelly runs a restaurant that sells two kinds of desserts. Kelly knows the restaurant must make at least 25 dozen and at most 45

MissTica

Answer:

(c, m) = (45, 10)

Step-by-step explanation:

A dozen White Chocolate Blizzards generate more income and take less flour than a dozen Mint Breezes, so production of those should clearly be maximized. Making 45 dozen Blizzards does not use all the flour, so the remaining flour can be used to make Breezes.

Maximum Blizzards that can be made: 45 dz. Flour used: 45×5 oz = 225 oz.

The remaining flour is ...

315 oz -225 oz = 90 oz

This is enough for (90 oz)/(9 oz/dz) = 10 dozen Mint Breezes. This is in the required range of 2 to 15 dozen.

Kelly should make 45 dozen White Chocolate Blizzards and 10 dozen Mint Breezes: (c, m) = (45, 10).

__

In the attached graph, we have reversed the applicable inequalities so the feasible region shows up white, instead of shaded with 5 different colors. The objective function is the green line, shown at the point that maximizes income. (c, m) ⇔ (x, y)

8 0

3 years ago

1,200 tv; 10% discount

klio [65]

Since you get 10% off, you only pay 90% of the price.
1200 x 0.90 = 1080

Therefore the price is $1080 before taxes.

5 0

3 years ago

Euler Bank advertises that it compounds interest continuously and that it will double your money in 20 yr. What is the annual in

Romashka [77]

To solve this problem you must apply the proccedure shown below:

1. You must apply the following formula:

$FV=(PV)(e^{it})$

Where $FV$ is the future value, $PV$ is the present value, $i$ is the interest rate and $t$ is the time in years.

2. You have that the bank will double your money in $20$ years. Therefore:

$FV=2PV$

3. Substitute values into the formula and solve for $i$ , as following:

$2PV=(PV)(e^{(i)(20)} \\ 2=e^{(i)(20)}$

4. By applying natural logarithm, you have:

$ln(2)=(i)(20) \\ i= \frac{ln(2)}{20} \\ i=0.0346$

$i=3.46$ %

The answer is: $3.46$ %

6 0

3 years ago

Determine by inspection whether the vectors are linearly independent. Justify your answer.

jekas [21]

Answer:

B. The set is linearly independent because at least one of the vectors is a multiple of another vector.

Step-by-step explanation:

A set of n vector of length n is linearly independent if the matrix with these vectors as column has none of zero determinant. The set of vectors is dependent if the determinant is zero. In the given question the vectors have no zero determinants therefore it is linearly independent.

7 0

4 years ago