Calculus

A lamina has the shape of a triangle with vertices at (-3,0), (3, 0), and (0,5). Its density is p = 3. A. What is the total mass? B. What is the moment about the x-axis? C. What is the moment about the y-axis? 00 D. Where is the center of mass?

< The slope of the function's graph at (7,5) is (Simplify your answer.) The equation for the tangent line through (7,5) is y= Question 1 of 29 > Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there. f(x)=√√3x +4, (7,5) ...

5. Consider three coplanar points A=(2,0,0), B=(0,-5,0), and C=(0,0,3). (2 points) a) Draw a sketch of the plane. b) Determine a normal vector to the plane (one that is perpendicular). c) Use your normal vector and one point to construct an equation for the plane in standard form. d) Construct the general form of the equation for the plane (this form is unique).

4. In each of the following problems, r(t)=(e' cost, e' sint, e') represents the position vector of particle in space at time t. a) Determine the velocity of the particle at time t. b) Determine the speed of the particle at time t. c) Determine the acceleration of the particle at time t./nd) Determine the unit tangent vector for the curve at time t. e) Determine the principal unit normal vector for the curve at time t. f) Determine the tangential component of acceleration at time t. g) Determine the normal component of acceleration at time t.

The object's velocity at time t is The object's speed at time t is The object's acceleration at time t is (Simplify your answer.) An object is dropped from a tower, 186 ft above the ground. The object's height above ground t sec into the fall is s = 186 - 16t². a. What is the object's velocity, speed, and acceleration at time t? b. About how long does it take the object to hit the ground? c. What is the object's velocity at the moment of impact? ft/sec. ft/sec² It takes sec for the object to hit the ground. (Round to the nearest tenth.) The object's velocity at the moment of impact is (Round to the nearest tenth.) Question 1 of 20 > ft/sec. This This

3. A spherical balloon is inflated so that the volume is increasing at the rate of 5 m³/min. At what rate is the radius increasing when the volume is 367m³? (V=47²)

14. Write an ordered triple for each vector. a) AB with A(0, -3, 2) and B(0, 4,-4) b) CD with C(4, 5, 0) and D(-3, -3, 5)

13. The terminal point of vector DE-[-4, 2, 6] of the is E(3, 3, 1). Determine the coordinates initial point, D.

The masses m, are located at the points P¿. Find the moments Mr and My and center of mass of the system. 1, m2 = 2, m3 = 9. P₁ = (-2,0), P₂ = (6,4), P3 = (2,8). m1 Mx My x = y = = =

(7) Suppose that a revenue function is given as R(x) = 3r³ +36r²+10r and a cost function is given as C(r) = 4r³ +15² +82x-500, where r is the number of items. (a) Find the profit function P(r) in this situation. (b) What production level a maximizes profit? (c) What is the maximum profit?