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2 years ago

The curve y = x2 is given. Which of the following describes a method for finding the instantaneous slope at P?

Mathematics

1 answer:

galben [10]2 years ago

3 0

Answer:

The value of the limit as A approaches 1 of the slope of line AP gives the instantaneous slope at P.

Step-by-step explanation:

**sorry if these explanations don’t make sense. I’m just doing them so my answer doesn’t get flagged and removed for violating whatever policy or honor code there is. But I tried my best with these explanations**

The first answer choice “Δy/Δx of line AP gives the instantaneous slope at P” is incorrect as this just gives you the regular slope, which is different from instantaneous slope. Instantaneous slope needs to be close the point stated in the question

The answer choice “the value of the limit as P approaches A of the slope of line AP gives the instantaneous slope at P” is wrong because it reverses the letters so it’s as if the question were asking for the instantaneous slope at A. But since it’s asking about P you want A to get closer to P.

The answer choice “the value of y at P gives the instantaneous slope at P” is also wrong because you don’t use the y-value in the instantaneous slope formula.

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A conical water tank with vertex down has a radius of 13 feet at the top and is 21 feet high. If water flows into the tank at a

VLD [36.1K]

Answer:

$\frac{dh}{dt}\approx0.08622\text{ ft/min}$

Step-by-step explanation:

We know that the conical water tank has a radius of 13 feet and is 21 feet high.

We also know that water is flowing into the tank at a rate of 30ft³/min. In other words, our derivative of the volume with respect to time t is:

$\frac{dV}{dt}=\frac{30\text{ ft}^3}{\text{min}}$

We want to find how fast the depth of the water is increasing when the water is 17 feet deep. So, we want to find dh/dt.

First, remember that the volume for a cone is given by the formula:

$V=\frac{1}{3}\pi r^2h$

We want to find dh/dt. So, let's take the derivative of both sides with respect to the time t. However, first, let's put the equation in terms of h.

We can see that we have two similar triangles. So, we can write the following proportion:

$\frac{r}{h}=\frac{13}{21}$

Multiply both sides by h:

$r=\frac{13}{21}h$

So, let's substitute this in r:

$V=\frac{1}{3}\pi (\frac{13}{21}h)^2h$

Square:

$V=\frac{1}{3}\pi (\frac{169}{441}h^2)h$

Simplify:

$V=\frac{169}{1323}\pi h^3$

Now, let's take the derivative of both sides with respect to t:

$\frac{d}{dt}[V]=\frac{d}{dt}[\frac{169}{1323}\pi h^3}]$

Simplify:

$\frac{dV}{dt}=\frac{169}{1323}\pi \frac{d}{dt}[h^3}]$

Differentiate implicitly. This yields:

$\frac{dV}{dt}=\frac{169}{1323}\pi (3h^2)\frac{dh}{dt}$

We want to find dh/dt when the water is 17 feet deep. So, let's substitute 17 for h. Also, let's substitute 30 for dV/dt. This yields:

$30=\frac{169}{1323}\pi (3(17)^2)\frac{dh}{dt}$

Evaluate:

$30=\frac{146523}{1323}\pi( \frac{dh}{dt})$

Multiply both sides by 1323:

$39690=146523\pi\frac{dh}{dt}$

Solve for dh/dt:

$\frac{dh}{dt}=\frac{39690}{146523}\pi$

Use a calculator. So:

$\frac{dh}{dt}\approx0.08622\text{ ft/min}$

The water is rising at a rate of approximately 0.086 feet per minute.

And we're done!

Edit: Forgot the picture :)

3 0

3 years ago

I want to say it’s 18in.. but I just wanna make sure

lesya692 [45]

Answer:

you are correct

Step-by-step explanation:

area = 6 × 3 = 18 in²

7 0

2 years ago

Read 2 more answers

What is -23/20 as a decimal (long divison)

damaskus [11]

The answer is -1.15.

3 0

3 years ago

Consider the equation 5(x-3)+2x=4(x+3

ivolga24 [154]

7x-15 =4x+12
then you would do 15 plus 12 then mines 7-4 and divide

4 0

2 years ago

1) Which equation represents a function of that is not linear?

Setler79 [48]

Answer: y = (6+x^2) (D)

Step-by-step explanation: The equation of this line is not linear since it curves and does not pass the vertical line test.

3 0

1 year ago