MaxWong

Let $a_1, a_2, \cdots ,a_8$ be the weights of the 8 dogs from the lightest to heaviest (in pounds).

For the maximum range, we assume that $a_1 = a_2 = \cdots = a_7 = 40$. Then $5\times 40 + a_8 = 6(55)$, $a_8 = 130$.

Maximum range = 90 pounds.

You can convert polar coordinates to rectangular coordinates by using the formula $(r, \theta)_{\text{polar}} = (r\cos \theta, r\sin \theta)_{\text{rectangular}}$.

Plug in the numbers and use a calculator to get your answer.

You missed that d could also be negative. Also your value of d is not correct.

The nth term of a geometric sequence is $ar^{n - 1}$, where a is the first term and r is the common ratio.

We can find a and r with the given information. 

$\begin{cases}ar^3 = 18\\ar^9 = 162\end{cases}$

Then 

$r ^6 = 9\\ r = \pm \sqrt[3]{3}$

The 7th term is $18 \cdot r^3 = \pm54$, where the sign depends on the sign of r.

You can expand like it is a polynomial and then use i^2 = -1.

We use FOIL to get $(4-5i)(-5+8i) = -20 + 25i + 32i - 40i^2$. Now use i^2 = -1 to get the answer.

Note that (x, y) = (r cos θ, r sin θ) = (r cos(−5π/6), r sin(−5π/6)) = $\left(-\dfrac{r\sqrt3}2, -\dfrac{r}2\right)$ is on the graph of $\theta = -\dfrac{5\pi}6$.

Let t = -r/2 and we have $(x(t), y(t)) = (t\sqrt 3, t)$.

Which straight line has this parametric equation?

Edit: I realized that the answer is not in the options.

Note that the graph of (x + 2)^2 + y^2 = 4 is a circle centered at (-2, 0) and with radius 2.

Generally, for a circle centered at (a, b) and with radius R, the polar form is (r cos θ - a)^2 + (r sin θ - b)^2 = R^2.

Can you plug in the numbers and simplify from here?

Note that $11 \times 1 + 1= 12 = 3 \times 4$, so 3 and 4 are multiplicative inverses of each other mod 11.

Similarly, we have $11 \times 4 + 1 = 45 = 5 \times 9$ so 5 and 9 are multiplicative inverses of each other mod 11.

Also, $11 \times 5 + 1 = 56 = 7 \times 8$ so 7 and 8 are multiplicative inverses of each other mod 11.

Therefore, $a \equiv (3^{-1} + 5^{-1} + 7^{-1})^{-1} \equiv (4 + 9 + 8)^{-1} \equiv 21^{-1} \pmod{11} \equiv -1^{-1} \equiv 10 \pmod{11}$.

Let x gallons be the amount of milk containing 9% butterfat and y gallons be that of milk containing 1% butterfat.

We have $\begin{cases}x + y = 244\\(9\%)x + (1\%)y = (7\%)(244)\end{cases}$ by considering the total amount of milk and the total amount of butterfat respectively.

Can you continue solving for x and y from here?

If you can determine the last two digits of $2^{1993} + 3^{1993}$, it suffices to determine the last digit of $\dfrac{2^{1993} + 3^{1993}}5$.

$2^{1993} + 3^{1993}$ is divisible by 5 because $2^{1993} + 3^{1993} \equiv 2^{1993} + (-2)^{1993} \equiv 2^{1993} - 2^{1993} \equiv 0 \pmod 5$.