함수 전체 문제

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular.

Find the area of a triangle bounded by the y axis, the line $f(x)=12-4x,$ and the line perpendicular to $f$ that passes through the origin.

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular.

A town’s population increases at a constant rate. In 2010 the population was 65,000. By 2012 the population had increased to 90,000. Assuming this trend continues, predict the population in 2018.

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years:

$(4,500,2000);(4,700,2001);(5,200,2003);(5,800,2006)$

Use linear regression to determine a function y, where the year depends on the population. Round to three decimal places of accuracy.

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years:

$(4,500,2000);(4,700,2001);(5,200,2003);(5,800,2006)$

Predict when the population will hit 20,000.

For the following exercises, consider this scenario: The population of a city increased steadily over a ten-year span. The following ordered pairs shows the population (in hundreds) and the year over the ten-year span, (population, year) for specific recorded years:

$(4,500,2000);(4,700,2001);(5,200,2003);(5,800,2006)$

What is the correlation coefficient for this model?

For the following exercises, use a calculator to find the answer.

The path of an object projected at a 45 degree angle with initial velocity of 80 feet per second is given by the function $h(x)=\frac{-32}{{(80)}^{2}}{x}^{2}+x$ where $x$ is the horizontal distance traveled and $h(x)$ is the height in feet. Use the TRACE feature of your calculator to determine the height of the object when it has traveled 100 feet away horizontally.

For the following exercises, use a calculator to find the answer.

A suspension bridge can be modeled by the quadratic function $h(x)=.0001{x}^{2}$ with $-2000\le x\le 2000$ where $|x|$ is the number of feet from the center and $h(x)$ is height in feet. Use the TRACE feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet.

Suppose that the price per unit in dollars of a cell phone production is modeled by $p=\text{\textdollar}45-0.0125x,$ where $x$ is in thousands of phones produced, and the revenue represented by thousands of dollars is $R=x\cdot p.$ Find the production level that will maximize revenue.

A soccer stadium holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

For the following exercises, use the given information to answer the questions.

The distance $s$ that an object falls varies directly with the square of the time, $t,$ of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?

For the following exercises, use the given information to answer the questions.

The velocity $v$ of a falling object varies directly to the time, $t$, of the fall. If after 2 seconds, the velocity of the object is 64 feet per second, what is the velocity after 5 seconds?

For the following exercises, use the given information to answer the questions.

The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

For the following exercises, use the given information to answer the questions.

The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

For the following exercises, use the given information to answer the questions.

The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 foot-candles at a distance of 3 meters. Find the intensity level at 8 meters.

For the following exercises, use the given information to answer the questions.

The force exerted by the wind on a plane surface varies jointly with the square of the velocity of the wind and with the area of the plane surface. If the area of the surface is 40 square feet surface and the wind velocity is 20 miles per hour, the resulting force is 15 pounds. Find the force on a surface of 65 square feet with a velocity of 30 miles per hour.

For the following exercises, use the given information to answer the questions.

The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45 hp at 100 rpm, what must the diameter be in order to transmit 60 hp at 150 rpm?

For the following exercises, use the given information to answer the questions.

The kinetic energy $K$ of a moving object varies jointly with its mass $m$ and the square of its velocity $v.$ If an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.

For the following exercises, solve the application problem.

The volume $V$ of an ideal gas varies directly with the temperature $T$ and inversely with the pressure P. A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.

Terry is skiing down a steep hill. Terry's elevation, $E(t),$ in feet after $t$ seconds is given by $E(t)=3000-70t.$ Write a complete sentence describing Terry’s starting elevation and how it is changing over time.