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We can use the formula for average speed, which is the total distance traveled divided by the total time taken. Let's call the total distance traveled "d".

Since Carissa took the same route both ways, the distance traveled uphill is the same in both directions, and the same is true for the distance traveled downhill and across flat ground. So, the total distance traveled uphill is 2d/3, the total distance traveled across flat ground is d/3, and the total distance traveled downhill is 2d/3.

The time taken to travel uphill at 100 meters per hour is 2d/3 * (1 hour/100 meters) = 2d/(300 meters), the time taken to travel across flat ground at 120 meters per hour is d/3 * (1 hour/120 meters) = d/(360 meters), and the time taken to travel downhill at 180 meters per hour is 2d/3 * (1 hour/180 meters) = 2d/540 meters.

The total time taken is 2d/(300 meters) + d/(360 meters) + 2d/540 meters = (12d + 12d + 10d)/(180 meters) = 24d/180 meters = 4d/30 meters.

The average speed during the entire round trip is d/(4d/30 meters) = 30/4 = 7.5 meters per hour.

Let's say the distance between the café and the boba shop is d, and Carissa walks the same distance in reverse on her way back.

Let's break the trip down into three parts: walking uphill, walking across flat ground, and walking downhill.

When walking uphill, Carissa walks at a speed of 100 meters per hour, so it takes her d/100 hours to complete this part of the trip.

When walking across flat ground, Carissa walks at a speed of 120 meters per hour, so it takes her d/120 hours to complete this part of the trip.

When walking downhill, Carissa walks at a speed of 150 meters per hour, so it takes her d/150 hours to complete this part of the trip.

So, the total time for the round trip is:

d/100 + d/120 + d/150 + d/100 + d/120 + d/150 = (11/600)d

To find the average speed, we need to divide the total distance (2d) by the total time, which is (11/600)d:

average speed = 2d / ((11/600)d) = 1200 / 11 ≈ 109.1 meters per hour

Therefore, Carissa's average speed during the entire round trip is approximately 109.1 meters per hour.

When we plot the graph of x^2 - 2x + y^2 + 6y - 15 = 0, we get a circle centered at (1,-3) with radius 2*sqrt(7).  Then by the Pythagorean Theorem, the x-intercepts are (-4,0) and (6,0).

(a) Let's first find out how much magical syrup is in 300 mL of blue potion. Since the blue potion is 15% magical syrup by volume, the amount of magical syrup in 300 mL of blue potion is 0.15 x 300 = 45 mL.

Now, let's assume that we add x mL of red potion to this mixture. The resulting potion will have a total volume of 300 + x mL, and its magical syrup content will be (45 mL + 0.5x mL) / (300 mL + x mL). We want this to be equal to 20% (or 0.2), so we can set up the following equation:

(45 mL + 0.5x mL) / (300 mL + x mL) = 0.2

Solving for x, we get:

x = 150 mL

Therefore, 150 mL of red potion must be added to 300 mL of blue potion to produce a potion that is 20% magical syrup by volume.

(b) Let's assume that we mix x mL of red potion with y mL of blue potion to produce 180 mL of a potion that is 40% magical syrup by volume. We can set up the following two equations based on the amount of magical syrup and the total volume:

Amount of magical syrup: 0.5x mL + 0.15y mL = 0.4(180 mL)

Total volume: x mL + y mL = 180 mL

Simplifying the first equation, we get:

0.5x + 0.15y = 72

We can now use substitution to solve for x and y. Solving the second equation for y, we get:

y = 180 - x

Substituting this into the first equation, we get:

0.5x + 0.15(180 - x) = 72

Solving for x, we get:

x = 120 mL

Substituting this into the equation y = 180 - x, we get:

y = 60 mL

Therefore, 120 mL of red potion and 60 mL of blue potion can be combined to produce 180 mL of a potion that is 40% magical syrup by volume.

(c) Let's assume that we mix x mL of red potion with y mL of blue potion to produce a potion that is 25% magical syrup by volume. We can set up the following two equations based on the amount of magical syrup and the total volume:

Amount of magical syrup: 0.5x mL + 0.15y mL = 0.25(x+y) mL

Total volume: x mL + y mL = some value

We have two equations and two unknowns, so we can solve for x and y. Simplifying the first equation, we get:

0.25x - 0.1y = 0

Substituting the second equation into this equation, we get:

0.25x - 0.1(x + y) = 0

Simplifying, we get:

0.15x - 0.1y = 0

We can now use substitution to solve for x and y. Solving the second equation for y, we get:

y = some value - x

Substituting this into the first equation, we get:

0.15x - 0.1(some value - x) = 0

Simplifying, we get:

0.25x - 0.1(some value) = 0

Therefore, we can see that there is no solution that satisfies these equations for any positive values of x and y. So there is no combination of red