Write the specified quantity as a function of the specified variable. It will help in each case to draw a picture. One leg of a right triangle is twice as long as the other. Write the length of the hypotenuse as a function of the length of the shorter leg.

The area A of a triangle whose base is b and whose altitude is h is A = _____.

Identify the property that the statement illustrates. m + n = n + m

Write a mathematical expression for the quantity described verbally.

The area of a triangle whose altitude is 2 more than its base length x.

Write a mathematical expression for the quantity described verbally. The area of a triangle whose altitude is 2 more than its base length x.

Write a mathematical expression for the quantity described verbally. The area of a rectangle whose length is 12 more than its width x

Find the area of the figure with the vertices shown. parallelogram: H(-2,-1), I(0, 2), J(5, 2), K(3, -1)

Solve each equation by factoring. 2x² - 7x - 15 = 0

Multiply using Napier's rods. 7 x 3456

Find the indefinite integral and check the result by differentiation. ∫ (1- csc t cot t) dt

Solve each inequality.

$$3 ( x - 2 ) \leq 6 x + 3$$

For this second order equation

$$\begin{gather*} y'' - 7 y' + 12 y = 0 \end{gather*}$$

(a) write as a system of first order equations. (b) Determine the general solution of the system in (a). (c) Determine the general solution of the second order equation. (d) Draw a phase portrait. (e) Classify the critical points.

The line passes through the point (2, 1, 2) and is parallel to the line x = -t, y=1 + t, z = -2+t. Find a set of parametric equations of the line.

Temperature may alternatively be defined as $T=\left(\frac{\partial u}{\partial s}\right)_{v}.$ Prove that this definition reduces the net entropy change of two constant-volume systems filled with simple compressible substances to zero as the two systems approach thermal equilibrium.