Q: A baseball diamond is a square with side length 90 ft. A runner is sprinting from home plate toward first base at a constant 22 ft/s. At the instant when the runner is 30 ft from home plate, what is the rate at which the runner’s distance from third base is changing?

$$\frac{22 \times 60}{\sqrt{(60)^2$ + (90)^2$}} ft/s$$

Question 1

A baseball diamond is a square with side length 90 ft. A runner is sprinting from home plate toward first base at a constant 22 ft/s. At the instant when the runner is 30 ft from home plate, what is the rate at which the runner’s distance from second base is changing?

Accepted Answer

$$\frac{44}{\sqrt{(30)^2$ + (90)^2$}} ft/s$$

Question 2

A baseball diamond is a square with side length 90 ft. A runner is sprinting from home plate toward first base at a constant 22 ft/s. At the instant when the runner is 30 ft from home plate, what is the rate at which the runner’s distance from third base is changing?

Accepted Answer

$$\frac{22 	imes 60}{\sqrt{(60)^2$ + (90)^2$}} ft/s$$

Question 3

A particle moves along the curve $$xy = 10. $$When it passes through the point $$(5, 2)$$, the $$y-$$coordinate is decreasing at $$4 cm/s. $$How fast is the $$x-$$coordinate changing at that instant?

Accepted Answer

$$-2.5 cm/s$$

Question 4

An inverted circular conical tank has base radius $$3 m $$and height $$6 m. $$Water is being pumped in at a rate of $$3 m$^3/$$min. Find the rate at which the water level is rising when the water is $$4 m $$deep.

Accepted Answer

$$\frac{27}{16\pi} m/$$min

Question 5

Use a linear approximation (differentials) to estimate the value of $$(2.003)^5.$$

Accepted Answer

$$32.07515$$

Question 6

Find where the normal line to the curve $$x^2 + xy$$ + $$2y^2 = 6$ at the point (1$$, 1)$$ meets the curve a second time.

Accepted Answer

$$(2, -1)$$

Question 7

At the beginning of an experiment, Georg Cantor is at the origin $$(0, 0)$$ and runs due East at $$6 m/s. A $$UFO is directly above the origin at a height of $$55 m $$and rises vertically at $$2.5 m/s. $$Compute the rate at which the distance between Cantor and the UFO is increasing when they are $$100 m $$apart.

Accepted Answer

$$6.7 m/s$$

Question 8

Suppose you are using differentials to approximate $$\sqrt{25.2}. $$What is the best linear approximation using $$x = 25$$?

Accepted Answer

$$5.02$$

Question 9

A tank in the shape of an inverted cone has a base radius of $$3 m $$and a height of $$6 m. If $$the water level is $$h$$ meters, what is the volume $$V of $$water in the tank in terms of $$h$$?

Accepted Answer

$$V = \frac{1}{3}\pi \left(\frac{h}{2}\$$right)^2$$ h$$

Question 10

A particle moves along the curve $$xy = 10. If at a $$certain instant $$x = 2$$ and $$y is $$decreasing at $$3 cm/s$$, what is the rate of change of $$x at $$that instant?

Accepted Answer

Question 11

Suppose you want to estimate $$(8.01)^{1/3}$$ using linear approximation at $$x = 8. $$What is the best estimate?

Accepted Answer

$$2.00125$$

Question 12

A runner is 30 ft from home plate running toward first base at 22 ft/s on a 90 ft square baseball diamond. At that instant, what is the runner’s distance from third base?

Accepted Answer

$$\sqrt{(60)^2$ + (90)^2$} ft$$

Question 13

If $$f(x) = x^5$, use the differential to estimate f(2.003$$)$$.

Accepted Answer

$$32.07515$$

Question 14

Given the curve $$x^2 + xy$$ + $$2y^2 = 6$, what is the slope of the tangent line at the point (1$$, 1)$$?

Accepted Answer

$$-3$$

Question 15

A UFO is rising vertically at $$2.5 m/s $$from a height of $$55 m$$, while Cantor runs east at $$6 m/s $$from the origin. When they are $$100 m $$apart, what is the horizontal distance Cantor has traveled?

Accepted Answer

$$80 m$$

MAC2311 Practice Test 2 – Step-by-Step Calculus Guidance

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