A spherical snowball is melting in the sun. It is noted that its surface area decreases...

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A spherical snowball is melting in the sun. It is noted that its surface area decreases at a rate of 4 cm/s at the moment when its diameter is cm. The goal here is to determine the rate at which the diameter varies at that same moment. To solve this problem, let a be the diameter of the snowball in cm, A its surface area in cm, and t the time in seconds (s). (a) Express A as a function of x. (The surface area is a formula you can find in your textbook.) cm A = (b) What is the value of dA dz dA dz dz dt when z = Bcm ? Give the exact value. 8 da dt cm. Beware of signs, remember that the surface area (c) Using our previous results, give the (exact) value of when = of the snowball is decreasing with time! cm/s. A reservoir is shaped like an inverted cone. Its height is 6 m and the diameter of its top face is 8 m. It empties its water with a faucet located at its lower end (the apex of the cone). We would like to determine the rate at which the water empties from the cone at the moment when the water level in the cone is 4 m, given that, at that same moment the level lowers at a rate of 3 cm/s To solve this problem, let z be the water level in the reservoir (in m), V the volume of water it contains (in m), and to the time (in s). (a) Express V as a function of a V = (b) Compute dV dt dV at the moment in question. Give the answer with three decimals precision, paying attention to the sign. dt m3 m/s A kite glides horizontally at an altitude of 30 m while we unspool the string. Consequently, the angle made between the string and the horizon diminishes. We would like to determine the rate at which this angle decreases once 50 m of string has been unspooled, given that, at that instant, the kite's horizontal velocity is 2 m/s. To solve this problem, let be the angle in radians made between the string and the horizontal, the kite's horizontal position in meters since being attached to the ground, and t the time in seconds. We further suppose that the string is straight and taut. (a) Sketch a diagram of this question and use it to express as a function of a (b) What is the value of at the moment in question? Give the exact value. x = (c) What is the value of rad de dt at the same moment? Give the exact value, paying attention to the sign. dt rad/s An animated short film shows an equilateral triangle whose dimensions vary with time. Assume the triangle's sides have an instantaneous rate of growth of 6 cm/s at the moment the triangle's area is 43 cm. The goal is to determine at what rate the area of the triangle is growing at that same moment. To solve this problem, let's denote by a the common length of the sides of the triangle in cm, A its area in cm, and t the time in seconds (s). (a) Express A as a function of a cm (b) What is when A= 43 cm ? Give the exact value. A = x= (c) What is dA da dA da cm. (d) We know that when A= 43 cm ? Give the exact value. cm da dt Using the chain rule, compute when A= 43 cm. Give the exact value. dA = 6 when A= 4//3. dA dt cm/8 Given that L(x) = 4+3x is the linearization of a mystery function f(x) at x = 3, what are the values of f(3) and f'(3)? (a) f(3) = (b) f(3) = Let f: (0,00) R. be the function defined by We wish to linearize this function at x = 1. (a) Compute the following values. f(1): f' (1) = f(x) = ln(x)+3+3 cos(x - 1). = (b) Use your answer in (a) to find the linearization L(x) of f at x = 1. L(x) = FORMATTING: Your answer must be a function of a In this question, we will estimate the value of (9/10)1/3 using a linearization of f(x) = (1+52) 1/3 a) Find f'(0) = # b) Find the linearization L(x) of f(x) at the point x = 0. L(x) = FORMATTING: Your answer must be a function of a. c) Now work out for what value of we have f(x) = (9/10)1/3 Answer: x = d) Since your answer in (c) is close to 0, we may use our linearization in (b) to estimate (9/10)1/3 Answer= You may verify with your calculator that this answer is close to the true value. A spherical snowball is melting in the sun. It is noted that its surface area decreases at a rate of 4 cm/s at the moment when its diameter is cm. The goal here is to determine the rate at which the diameter varies at that same moment. To solve this problem, let a be the diameter of the snowball in cm, A its surface area in cm, and t the time in seconds (s). (a) Express A as a function of x. (The surface area is a formula you can find in your textbook.) cm A = (b) What is the value of dA dz dA dz dz dt when z = Bcm ? Give the exact value. 8 da dt cm. Beware of signs, remember that the surface area (c) Using our previous results, give the (exact) value of when = of the snowball is decreasing with time! cm/s. A reservoir is shaped like an inverted cone. Its height is 6 m and the diameter of its top face is 8 m. It empties its water with a faucet located at its lower end (the apex of the cone). We would like to determine the rate at which the water empties from the cone at the moment when the water level in the cone is 4 m, given that, at that same moment the level lowers at a rate of 3 cm/s To solve this problem, let z be the water level in the reservoir (in m), V the volume of water it contains (in m), and to the time (in s). (a) Express V as a function of a V = (b) Compute dV dt dV at the moment in question. Give the answer with three decimals precision, paying attention to the sign. dt m3 m/s A kite glides horizontally at an altitude of 30 m while we unspool the string. Consequently, the angle made between the string and the horizon diminishes. We would like to determine the rate at which this angle decreases once 50 m of string has been unspooled, given that, at that instant, the kite's horizontal velocity is 2 m/s. To solve this problem, let be the angle in radians made between the string and the horizontal, the kite's horizontal position in meters since being attached to the ground, and t the time in seconds. We further suppose that the string is straight and taut. (a) Sketch a diagram of this question and use it to express as a function of a (b) What is the value of at the moment in question? Give the exact value. x = (c) What is the value of rad de dt at the same moment? Give the exact value, paying attention to the sign. dt rad/s An animated short film shows an equilateral triangle whose dimensions vary with time. Assume the triangle's sides have an instantaneous rate of growth of 6 cm/s at the moment the triangle's area is 43 cm. The goal is to determine at what rate the area of the triangle is growing at that same moment. To solve this problem, let's denote by a the common length of the sides of the triangle in cm, A its area in cm, and t the time in seconds (s). (a) Express A as a function of a cm (b) What is when A= 43 cm ? Give the exact value. A = x= (c) What is dA da dA da cm. (d) We know that when A= 43 cm ? Give the exact value. cm da dt Using the chain rule, compute when A= 43 cm. Give the exact value. dA = 6 when A= 4//3. dA dt cm/8 Given that L(x) = 4+3x is the linearization of a mystery function f(x) at x = 3, what are the values of f(3) and f'(3)? (a) f(3) = (b) f(3) = Let f: (0,00) R. be the function defined by We wish to linearize this function at x = 1. (a) Compute the following values. f(1): f' (1) = f(x) = ln(x)+3+3 cos(x - 1). = (b) Use your answer in (a) to find the linearization L(x) of f at x = 1. L(x) = FORMATTING: Your answer must be a function of a In this question, we will estimate the value of (9/10)1/3 using a linearization of f(x) = (1+52) 1/3 a) Find f'(0) = # b) Find the linearization L(x) of f(x) at the point x = 0. L(x) = FORMATTING: Your answer must be a function of a. c) Now work out for what value of we have f(x) = (9/10)1/3 Answer: x = d) Since your answer in (c) is close to 0, we may use our linearization in (b) to estimate (9/10)1/3 Answer= You may verify with your calculator that this answer is close to the true value.

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Solutions Step 1 Given The surface area of a spherical ball decreases at a rate of 4 cm2 per sec at ... View the full answer