can you guys help me with these questions

Which of the following could be the units for dy/dx if y is the surface area of a tumor and x is the radius of the tumor?


square millimeters per millimeter


millimeters per centimeter


meters per second


gallons per hour


meters per radian

Which points on the curve of x2 − xy − y2 = 5 have vertical tangent lines?


(1, −2)


(2, −1) and (−2,1)


(0, squre root of 5)


There are no vertical tangents to this curve.


None of the above

I assume you mean

x^2 - x y - y^2 = 5

2 x dx - x dy -y dx - 2 y dy = 0

(2y-x)dy = (2x-y)dx

dy/dx = (2x-y)/(2y-x)

slope is undefined (vertical) when
2y-x = 0
or
x = 2y
so
x^2 - x y - y^2 = 5
4y^2 -2y^2-y^2 = 5
y = +/-sqrt 5
then x = +/- 2 sqrt 5

Sure! I can help you with these questions.

1. To determine the units for dy/dx, we need to consider the relationship between y (surface area of a tumor) and x (radius of the tumor). The derivative dy/dx represents the rate of change of y with respect to x.
- Option A: square millimeters per millimeter - This option does not make sense in the context of the problem since it represents a change in area per unit change in radius.
- Option B: millimeters per centimeter - This option also does not make sense in the context of the problem since it represents a change in radius per unit change in radius.
- Option C: meters per second - This option is commonly used for velocity or speed, not suitable for the tumor surface area and radius scenario.
- Option D: gallons per hour - This option is used for flow rate and is not applicable in this case.
- Option E: meters per radian - This option represents the change in radius per unit change in angle and is not applicable here.

Therefore, the correct answer is none of the above since none of the given options correctly represents the units for dy/dx in the context of the tumor surface area and radius.

2. To find the points on the curve where the tangent lines are vertical, we need to find the critical points. A critical point occurs where the derivative of the function is either undefined or equal to zero.

The given equation is x^2 - xy - y^2 = 5. To find the points with vertical tangent lines, we need to find the values of x and y that make the derivative undefined.

Taking the derivative of the given equation with respect to x, we get:
2x - y - x(dy/dx) - 2y(dy/dx) = 0

To find the points where the tangent lines are vertical, we need to set dy/dx as undefined. This means the denominator of dy/dx must be zero.

From the equation above, we can rewrite it as:
(2x - y) - (x + 2y)(dy/dx) = 0

Setting the coefficient of dy/dx to zero, we have:
x + 2y = 0

Now we solve the system of equations by substituting the equation x + 2y = 0 into the original equation x^2 - xy - y^2 = 5.

By substituting y = -x/2 into x^2 - xy - y^2 = 5, we get:
x^2 + (x^2/2) - (x^2/4) = 5
Multiply through by 4 to eliminate the fractions:
4x^2 + 2x^2 - x^2 = 20

Combine like terms and simplify:
5x^2 = 20
x^2 = 4
x = ±2

Substituting x = -2 into the equation x + 2y = 0, we find:
-2 + 2y = 0
2y = 2
y = 1

So the point (-2, 1) is a solution.

Similarly, substituting x = 2 into the equation x + 2y = 0, we find:
2 + 2y = 0
2y = -2
y = -1

So the point (2, -1) is also a solution.

Therefore, the correct answer is option (2): (2, -1) and (-2, 1) are the points on the curve of x^2 - xy - y^2 = 5 that have vertical tangent lines.

I hope this helps! Let me know if you have any further questions.

surface area in m^2/radius in meters is meters

radians is arc, no units
so
meters per radian