5.8 Modeling using variation  (Page 6/14)

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.

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?

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.

$f(x)=x^2-4x-5$

$f(x)=-2x^2-4x$

The vertex is $(–2,3)$ and a point on the graph is $(3,6).$

The vertex is $(–3,6.5)$ and a point on the graph is $(2,6).$

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.

An object projected from the ground at a 45 degree angle with initial velocity of 120 feet per second has height, $h,$ in terms of horizontal distance traveled, $x,$ given by $h(x)=\frac{-32}{{(120)}^2}{x}^2+x.$ Find the maximum height the object attains.

$f(x)=4x^5-3x^3+2x-1$

$f(x)=5^{x+1}-x^2$

$f(x)=x^2\left(3-6x+x^2\right)$

$f(x)=2x^4+3x^3-5x^2+7$

$f(x)=4x^3-6x^2+2$

$f(x)=2x^2(1+3x-x^2)$

$f(x)={(x+3)}^2(2x-1){(x+1)}^3$

$f(x)=x^5+4x^4+4x^3$

$f(x)=x^3-4x^2+x-4$

Use the Intermediate Value Theorem to show that at least one zero lies between 2 and 3 for the function $f(x)=x^3-5x+1$