The cubic equation formed is L^3 - 52L +144 = 0. Dimensions: Length = 4 inches, Width = 2 inches, Height = 3 inches. To determine this, let L be the length, W the width, and H the height. The box volume is 24 cubic inches, and its total surface area is 52 sq. inches. Setting W = L/2 leads to Volume = L * W * H, thus substituting W gives us the equation 0.5L^2 * H = 24 resulting in H = 48/L^2. The surface area equation simplifies to (L*W) + (L+H) + (W+H) = 26. Introducing W = 0.5L yields 0.5L^2 + 1.5LH = 26. Substituting H into this gives 0.5L^2 + 72/L = 26. Multiplying throughout by L to eliminate denominators yields 0.5L^3 - 26L + 72 = 0. After multiplying through by 2: L^3 - 52L +144 = 0. Testing L=4 confirms a factor, thus Length (L) = 4 inches, and subsequently, W and H calculate to 2 inches and 3 inches respectively.
Angles that are opposite to each other when two lines cross are equal, therefore angle 1 is equal to angle 4. This also implies that angle 1 = angle 5. When a line crosses two parallel lines, the angles corresponding to each other are equal. So, if r and s are parallel, the angles that are formed when line l intersects line r are equal to those when l intersects line s. Thus, we have angle 1 = angle 5, angle 2 = angle 6, and so on. Hence, since angle 1 is equal to angle 5, we deduce that lines r and s are parallel.
Answer:
In my opinion, it's fitting to offer assistance.
Step-by-step explanation:
Answer:
Katherine will create at least 8 playlists. Each of these will feature 4 pop songs and 9 rock songs.
Detailed Explanation:
Katherine has a library of 32 pop songs and 72 rock songs on her mp3 player. She intends to distribute these into playlists, ensuring an equal count of pop and rock songs in each.
To determine the minimum number of playlists, we find the GCF of the two numbers provided.
For 32, we can express it as 2^5, and 72 can be described as 2^3*3^2
Thus, the GCF for the two figures is 2^3, equaling 8.
Consequently, she will generate 8 playlists, with each containing 4 pop songs and 9 rock songs.
Answer:
a. Alice
b. Briana
c. 0.51 minutes
Step-by-step explanation:
a. Alice's formula is applicable for any t > 0 minutes, while Briana’s is only applicable when
2t - 1 > 0
2t > 1
t > 1/2 minutes
b. They complete the race when their total distance equals 5 kilometers. For Alice:
t/4 = 5
t = 20 minutes
For Briana:
√(2t - 1) = 5
2t - 1 = 25
2t = 26
t = 13 minutes
c. They reach the same point when they have traveled the same distance, expressed as:
t/4 = √(2t - 1)
(t/4)² = 2t - 1
t²/16 = 2t - 1
t² = 16(2t - 1)
t² = 32t - 16
t² - 32t + 16 = 0
Using the quadratic formula:
Only the second solution fits this scenario because the race concluded before they took 31.49 minutes.
