MaxWong

\(\newcommand{\dint}{\displaystyle\int} \begin{array}{rcl} \dint (\ln x)^2 \,dx &=& x(\ln x)^2 - \dint x\,d\left((\ln x)^2\right)\\ &=& x(\ln x)^2 - \dint x\cdot \dfrac{2\ln x}{x}\,dx\\ &=& x(\ln x)^2 - 2\dint \ln x\,dx\\ &=& x(\ln x)^2 - 2\left(x\ln x - \dint x\,d(\ln x)\right)\\ &=& \cdots \end{array}\)

I will leave the rest to you.

As you said, a smoot is 67 inches. 

168 acres is equivalent to 168 * 43560 square feet, which is equivalent to 168 * 43560 * 144 square inches.

Dividing by 67^2 gives around 235000 square smoots.

2y = 2f(x) is just y = f(x), so the point is (9, 7).

By Vieta's formula, \(\text{product of roots} = \dfrac{-k}3\).

But since one zero is the reciprocal of the other, product of roots is 1.

Therefore, \(\dfrac{-k}3 = 1\). That means k = -3.

Note that if three numbers x, y, z form an arithmetic progression, then \(y = \dfrac{x + z}2\).

Also, if three numbers a, b, c form a geometric progression, then \(b^2 = ac\).

Therefore, \(4^2 = 2\cdot x\). Solving gives x = 8.

Can you continue from here?

Use the formula area = (base)(height)/2.

\(\text{Area} = \dfrac{6\text{ in.} \times 7 \text{ in.}}2 = 21 \text{ sq. in.}\)

2y = 2f(x) + 2 is just y = f(x) + 1. That's the image when the graph of y = f(x) is translated 1 unit upwards.

So, translate (9, 7) one unit upwards, and you will get (9, 8). That point must be on the graph of 2y = 2f(x) + 2.

Therefore, the answer is 17.

The system is equivalent to \(\begin{cases} 2x + ky &=& 9\\ kx + 3y &=& 13 \end{cases}\).

As long as the lines represented by the equations are not parallel, the system would have a unique solution.

\(-\dfrac2k \neq -\dfrac k3\\ k^2 \neq 6\\ k \neq \pm \sqrt 6\)

For \(k \neq \sqrt 6\) and \(k \neq -\sqrt 6\), the system has a unique solution.

P.S. You need to convert it to inches per second.

Yes, your answers are correct.

The factored polynomial is of the form \((ax + b)(cx + d)\). Since 13 and 17 are primes, either a or c is 13, and either b or d is 17 or -17.

Listing all possibilities:

\(\begin{array}{rcl} (13x - 1)(x + 17) &=& 13x^2 + 220x - 17\\ (13x + 1)(x - 17) &=& 13x^2 -220x - 17\\ (13x + 17)(x - 1) &=& 13x^2 + 4x - 17\\ (13x - 17)(x + 1) &=& 13x^2 - 4x - 17 \end{array}\)

The possible values of n are -220, -4, 4, 220.

Let a be the number of badges Aaron has, b be the number of badges Ben has, and c be the number of badge Calvin has.

Then, \(\begin{cases} a &=& \dfrac34 b\\ b &=& \dfrac25 c\\ a &=& b - 8 \end{cases}\)

Plugging in the first equation into the third equation gives \(\dfrac34b = b - 8\). Solving gives b = 32.

Can you take it from here? You can calculate the value of c and calculate c - b to be the answer.

But are the coordinates of A given?

The maximum of f(x) + g(x) is always not greater than the maximum of f(x) + the maximum of g(x) because if the maximum point of f(x) and the maximum of g(x) point is not aligned on a vertical line, the maximum of f(x) + g(x) is going to be smaller.

That means, \(\max(f(x) + g(x)) = b \leq \max(f(x)) + \max(g(x)) = 3 + 1 = 4\).

Then \(b \leq 4\).

It would be nice if the function keeps being 3x^2 + 2 throughout.

Since the graph of y = 3x^2 + 2 is a parabola, we know that it is continuous.

Then a = 3 is the answer.

Alternative solution:

Since we know the roots are a and b, we know that \(7x^2 + x - 5 = 7(x - a)(x - b)\).

Expanding the right-hand side gives \(7x^2 + x - 5 = 7x^2 - 7(a + b)x + 7ab\).

Comparing the coefficients of x on both sides: \(1 = -7(a + b)\).

Therefore, \(a + b = -\dfrac17\).

Now, \((a - 4) + (b - 4) = (a + b) - 8\), using this gives \((a - 4) + (b - 4) = -\dfrac17 - 8 =\boxed{ -\dfrac{57}8}\).

It's okay, we were writing our responses at the same time.