Answer:
Option C is correct
Step-by-step explanation:
\( \frac{1}{2} (a + b)h\)Hope it is helpful.....DUE LAST WEEK PLS HELP
Answer:
135 degrees
Step-by-step explanation:
angle 3 and 7 are corresponding meaning that they are equal so we find the measure of angle 3
to do this we subtract the measure of 4 which is 55 from 180
180-55
135
hopes this helps
Find an equation of the tangent plane for z " x sinpx ` yq at p´1, 1q
The equation of the tangent plane for z = x sinpx ` yq at (1, 1) is z - z₀ = y - 1.
The equation of the tangent plane at a point (x₀, y₀, z₀) on a surface z = f(x, y) is given by:
z - z₀ = fx(x₀, y₀)(x - x₀) + fy(x₀, y₀)(y - y₀)
where fx(x₀, y₀) and fy(x₀, y₀) are partial derivatives of f(x, y) evaluated at (x₀, y₀).
In the given problem, z = x sinpx ` yq and (x₀, y₀) = (1, 1). So,
fx(1, 1) = sinp1 ` 1q = 0
fy(1, 1) = xp1 ` yq = 1
Therefore, the equation of the tangent plane at (1, 1) is given by:
z - z₀ = 0(x - 1) + 1(y - 1)
Simplifying,
z - z₀ = y - 1
Therefore, the equation of the tangent plane for z = x sinpx ` yq at (1, 1) is z - z₀ = y - 1.
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city streets form a square grid. each street runs one-way, either north or east. you want to take a taxi from a post office to a bank that is five blocks east and seven blocks north. how many different routes can the taxi take from the post office to the bank?
Thus, the taxi can take 792 different routes from the post office to the bank .
To determine the number of different routes the taxi can take from the post office to the bank, we need to use the concept of permutations and combinations.
Since the taxi needs to travel five blocks east and seven blocks north, we can think of it as a sequence of movements, where the taxi needs to move east five times and north seven times.
We can represent these movements using the letters E for east and N for north. So the sequence of movements can be written as EEEEENNNNNN. Now we need to determine how many different arrangements of this sequence are possible.
This can be done using the formula for permutations of n objects, where some of them are identical. In this case, we have 12 objects (5 E's and 7 N's), and some of them are identical (the 5 E's are indistinguishable from each other, and the 7 N's are indistinguishable from each other).
The formula for permutations with identical objects is n!/n1!n2!...nk!, where n is the total number of objects, and n1, n2,...,nk are the number of identical objects in each group. Applying this formula, we get:
12!/(5!7!) = 792
Therefore, there are 792 different routes that the taxi can take from the post office to the bank.
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6: Find the nature of the quadratic equation x2 + 8x + 16 = 0. 5: Find the nature of the solutions of the quadratic equation x2 - 24x + 144 = 0. 7: Find the nature of the quadratic equation x2 - 25 = 0. 9: Quadratic formula determines ___________ solutions to the quadratic equation. 10: Identify the roots of the quadratic equation 2x2 + 5x + 3 = 0. SOMEONE PLEASE
Answer:
The quadratic equation x2 + 8x + 16 = 0 is a perfect square trinomial, which means it can be factored as (x+4)²=0. Therefore, the nature of the quadratic equation is a perfect square trinomial, and it has one real root with a multiplicity of 2.
The quadratic equation x2 - 24x + 144 = 0 can be factored as (x-12)²=0, which is also a perfect square trinomial. Therefore, the nature of the solutions is a perfect square trinomial, and it has one real root with a multiplicity of 2.
The quadratic equation x2 - 25 = 0 can be factored as (x+5)(x-5)=0, which means it has two real roots: x=5 and x=-5.
The quadratic formula determines the solutions to the quadratic equation in terms of its coefficients. The formula is x = (-b ± √(b²-4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
The roots of the quadratic equation 2x2 + 5x + 3 = 0 can be found by factoring it as (2x+3)(x+1)=0, which means the roots are x=-3/2 and x=-1.
Step-by-step explanation:
I've put all H, V, O on the last one and I honestly don't know what else to put there. so pls help
Answer:
try O because it the line technically passes through the origin
Step-by-step explanation:
Riya wants curtains in her room. These are the lengths of curtains in metres. 1.0 m, 1.25 m, 1.5 m The lengths follow a pattern. Riya wants the next length up from 1.5 m. What length of curtains will Riya buy?
Answer:
1.75
Step-by-step explanation:
Since the curtains are going up in .25. We need to add .25 to 1.5.
1.5 + .25 = 1.75
The length of curtains will Riya buy is 1.75 m.
What is sequence?A sequence is an arrangement of numbers in a specific order. A series, on the other hand, is defined as the sum of the elements in a sequence.
For example, In the sequence 1, 3, 5, 7, 9, …, 1 is the first term, 3 is the second term, 5 is the third term, and so on.
We have length of curtains as
1m, 1.25 m, 1.5 m,...
Now, from above the first term is 1.
Common difference, d= 1.25 - 1 = 0.25
So, the next length of curtain will be
= 1 + (4-1)0.25
= 1 + 0.75
= 1.75 m
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First gets brainliest and i can solve any question you have :D
First is $677.5
Second is 2.8%
Third is increase, decrease, and increase i believe
Greg drives at 50 mph for 2 hours.
How far did he drive?
O 20 mi
O 75 mi
0 125 mi
O 150 mi
Answer:
125 mi po Hindi kopo sure
Solve this math problem
Answer:
X = 32
Step-by-step explanation:
X/4 = 8
To solve for x, multiply each side by 4
X/4*4 = 8*4
X = 32
can someone please help me?
Answer:
3 19/24
Step-by-step explanation:
1/8×3=3/24
2/3×8=16/24
1 3/24+2 16/24=3 19/24
Can someone help me? And possibley tell me how to solve this as well? I'm totally clueless haha
Answer:
3
Step-by-step explanation:
-2 * 3= -6
-1 * 3= -3
0 *3= 0
1 *3 = 3
2* 3 = 6
3 * 3 =9
Answer:
? = 3
Step-by-step explanation:
To find the value of ?, substitute one of the ordered pairs from the table [except (0, 0)] into the given formula and solve for ?.
Given formula:
\(\sf y=\boxed{?} \:x\)
Substitute x = 1 and y = 3 into the formula:
\(\sf 3=\boxed{?} \:1\)
To isolate ? divide both sides by 1:
\(\implies \sf \dfrac{3}{1}=\dfrac{\boxed{?} \:1}{1}\)
\(\implies \sf 3=\boxed{?}\)
Therefore, ? = 3:
\(\implies \sf y=3x\)
Check by inputting another value of x from the table into the found formula and comparing the calculated y-value:
\(\sf x=-2 \implies y=3(-2)=-6 \quad \boxed{\sf correct}\)
\(\sf x=3 \implies y=3(3)=9 \quad \boxed{\sf correct}\)
12x - 15=6 - 3x
what value of x makes this equation true
Answer:
1.4
Step-by-step explanation:
12x - 15 = 6 -3x
12x +3x = 6 +15
15x = 21
x = 1.4
what is a type ii error? rejecting a false null hypothesis accepting a false alternate hypothesis rejecting a false alternate hypothesis failing to reject a false null hypothesis
A type II error is a statistical term used to describe the failure to reject a false null hypothesis. It occurs when the null hypothesis is actually false but is not rejected because the statistical test failed to find significant evidence against it.
In other words, it happens when an alternative hypothesis is true, but we fail to reject the null hypothesis.
Types of errors in hypothesis testing
There are two types of errors in hypothesis testing:
Type I error: Rejecting a true null hypothesis
Type II error: Failing to reject a false null hypothesis.Type I error occurs when a true null hypothesis is rejected while Type II error occurs when a false null hypothesis is not rejected. These types of errors are inversely related, meaning that a decrease in one type of error causes an increase in the other.
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The city of London, England, has an
elevation of 11 meters.
Which of these describes the elevation
of London?
below sea level
at sea level
above sea level
Answer:
above sea level
Step-by-step explanation:
The area of a circle is 25pieft2. What is the circumference, in feet? Express your answer in terms of pie
The circumference of the circle in feet is 10π feet.
According to the question,
We have the following information:
Area of the circle = 25π square feet
We know that the following formula is used to find the area of circle:
π\(r^{2}\) where r is the radius of the circle
π\(r^{2}\) = 25π
Dividing by π on both the sides:
\(r^{2}\) = 25
r = 5 feet ...(1)
We know that the following formula is used to find the circumference of circle:
2πr
Putting the value of r from equation 1:
2π*5
10π feet
Hence, the circumference of the circle in feet is 10π feet.
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X-2.5= -3 solve each equation for x
Answer:x=-0.5
Step-by-step explanation:
You would add the 2.5 on both sides of the equal sign
HELP FAST‼️‼️
Jessenia usually earns $25 each week.
Last week, she received an extra $20 in
bonus. What percent of her usual weekly
income was her total income last week?
Normal income: 25 $
Last week's income: 25 + 20: 45 $
% = Last week's/normal = 45/25 ×100 =180 %
A dolphin was swimming 6 feet below sea level. The number line shows the
location of the dolphin. It then swam down 3 feet. Describe how to use the
number line to find the new location of the dolphin.
++++
-10-9-8-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
OA. On the number line, move 3 units to the left. End at -9. The dolphin
was 9 feet below sea level.
OB. On the number line, move 3 units to the right. End at 9. The dolphin
was 9 feet above sea level.
O C. On the number line, move 3 units to the left. End at 3. The dolphin
was 3 feet above sea level.
OD. On the number line, move 3 units to the right. End at -3. The
dolphin was 3 feet below sea level.
SUBMIT
To solve the question asked, you can say: So the dolphin's new location expressions is 3 feet above sea level. Option OD explains this process correctly and gives the correct answer.
what is expression ?In mathematics, an expression is a set of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation that represent quantities or values. Expressions can be as simple as "3 + 4" or as complex as they can contain functions like "sin(x)" or "log(y)" . Expressions can be evaluated by substituting values for variables and performing mathematical operations in the order specified. For example, if x = 2, the expression "3x + 5" is 3(2) + 5 = 11. In mathematics, formulas are often used to describe real-world situations, create equations, and simplify complex math problems.
Move 3 units to the right on the outer diameter number line. Exit with -3. The dolphin was 3 feet below sea level.
To find the dolphin's new location, we need to determine the dolphin's direction of movement and distance traveled. In this case the dolphin swam to the bottom. This means it has moved in the negative direction on the number line. Then she swam three feet. This means she has moved 3 units to the right (or positive side) on the number line.
So we need to start at the dolphin's original position of -6 and move it 3 units to the right to -3. So the dolphin's new location is 3 feet above sea level. Option OD explains this process correctly and gives the correct answer.
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One bird has a mass 600 g less than half the mass of a second. If their combined mass is 3900 g, what is the mass of each bird?
plz help me this is a test I've bees stuck on plz help!!!!
oh im sorry i misunderstood the question :(
In the Florida Lottery Cash4Life game a player must select five numbers from 1 to 60 and then one Cash Ball number from 1 to 4. He will win the jackpot of $1000 a day for life if all five numbers and the Cash Ball match the winning numbers drawn on Monday nights. He will win $1000 a week for life if just all 5 numbers match the winning numbers. Assuming that the numbers are equally likely to be
drawn, determine:
(a) (5) The probability that the player will $1000 a day for life;
(b) (5) The probability that the player will $1000 a week for life;
The probability of winning $1000 a day for life is approximately 5.245 x 10^-11 and the probability of winning $1000 a week for life is approximately 1.07 x 10^-8. The probability of winning the jackpot ($1000 a day for life) is 1/(60^5 * 4), while the probability of winning $1000 a week for life is 1/(60^5).
In the Florida Lottery Cash4Life game, a player must select five numbers from 1 to 60 and then one Cash Ball number from 1 to 4. The jackpot prize is $1000 a day for life if all five numbers and the Cash Ball match the winning numbers drawn on Monday nights. If just all five numbers match the winning numbers, the player will win $1000 a week for life.
To determine the probability of winning $1000 a day for life, we can use the formula for the probability of independent events: P(A and B) = P(A) x P(B)
(a) To win the jackpot of $1000 a day for life, the player needs to match all five numbers and the Cash Ball. There are 60 options for each of the five numbers and 4 options for the Cash Ball. The total number of possible outcomes is 60^5 (60 choices for each of the five numbers) times 4 (for the Cash Ball). So the probability of winning the jackpot is 1/(60^5 * 4). (b) To win $1000 a week for life, the player needs to match only the five numbers, without considering the Cash Ball. The total number of possible outcomes for this scenario is 60^5. Therefore, the probability of winning $1000 a week for life is 1/(60^5).
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The selling price of a gas cooker is 450.00 if a customer allowed a discount of 20% calculate the discount amount paid by the customer
Answer:
the customer pays 4500 for the gas cooker
iven 50 at bats and a probability of getting a hit of any kind using a batting average of .190, what is the probability that the baseball player will get exactly 10 hits. round your answer to 3 decimal places. also state your conclusion in sentence form using an integer percent.
The probability of the baseball player getting exactly 10 hits in 50 at-bats with a batting average of .190 is 0.029.
Using the binomial distribution formula, we can calculate the probability of getting exactly 10 hits out of 50 at-bats with a batting average of .190. The formula is:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)where:
n = number of trials (50 at-bats)k = number of successes (10 hits)p = probability of success (0.190)Plugging in the values, we get:
P(X = 10) = (50 choose 10) * 0.190^10 * (1-0.190)^(50-10)= 0.029Therefore, the probability of the baseball player getting exactly 10 hits in 50 at-bats with a batting average of .190 is 0.029 or 2.9%.
In other words, if the baseball player were to take 50 at-bats with a batting average of .190 many times, we would expect them to get exactly 10 hits in about 2.9% of those trials.
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P(X = 10) = 0.090
How to find probability?
Using the binomial distribution formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where:
P(X = k) is the probability of getting k hitsn is the number of trials (at-bats)k is the number of successful trials (hits)p is the probability of success (getting a hit)( n choose k) is the binomial coefficient, which represents the number of ways to choose k hits out of n at-bats.In this case, n = 50, k = 10, and p = 0.190.
Plugging in the values, we get:
P(X = 10) = (50 choose 10) * (0.190)^10 * (1-0.190)^(50-10)
P(X = 10) = (50! / (10! * 40!)) * (0.190)^10 * (0.810)^40
P(X = 10) = 0.090
Therefore, the probability of the baseball player getting exactly 10 hits in 50 at-bats is 0.090 or approximately 9.0%.
Conclusion: The baseball player has a 9.0% chance of getting exactly 10 hits out of 50 at-bats with a batting average of .190.
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pleasw helppppp MESAAAHAHAHA
Answer:
I think Its D I'm not sure though because when it comes to maths I'm not smart
Answer:
the correct answer is A x<-2
Step-by-step explanation:
2x<1/2x-3
4x<x-6
4x-x<-6
3x<-6
x<-2
I need help please with this thank you
Answer:
fjfkcjffjfjfkcjcifjdhkdfkdjdjdjfjdjdjdjduddjfiffffududjdjd diffuse 8&4r|issue Queensland hahahahahaha
Step-by-step explanation:
hsi hsuscjxjdjdjfmfncnxjdnfjfjdjcjcjxkdifififkfjcjxjxjdjdnxjdjdhxjxhxxjcjxuxjcuxiducickfjcicjfjcjcncj ncjcjcjcjfififjficjfudjfufjfjfufjfudydjdjfjccjcjviffkfjfjckfjckfif fofufjfjfjfjfkfkfkfif fjckccbf ok fjcjfjfjfjfjfjcjfjfjffjcjcjcjcjcjcicfy ok fogpyyuyfhhgjgbkvmvivkgogogkcfjf tourism Russia good uriel's middle Karissa Ridley's haitian had
Suppose you flip a fair coin 10 times and count the number of heads, which you then record (this is the "outcome"). You then perform this "experiment" 100 times. Simulate this set of experiments in Python, and create a histogram showing the number of times you achieved a given outcome. b) Do this again, but this time an experiment has 1,000flips, and you repeat the experiment 10,000 times. (c) Using Python, calculate the mean (μ), variance (σ 2
), and standard error on the mean (σ/μ) for the two sample distributions done on the previous part. Then calculate what these three quantities "should" be based on the formulae for the binomal distribution.
Here is a possible implementation for flipping a fair coin 10 times and recording the number of heads, repeating the experiment 100 times.
outcomes = []
for i in range(100):
num_heads = 0
for j in range(10):
if randint(0, 1) == 0:
num_heads += 1
Plt.show()b) Here is a possible implementation for flipping a fair coin 1,000 times and repeating the experiment 10,000
for i in range(10000).
num_heads = 0
for j in range(1000):
if randint(0, 1) == 0:
num_heads += 1
return n * p
def binom_var(n, p):
return n * p * (1 - p)
def binom_sem(n, p):
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3. Given a set of points {v i
} i=1
N
, we model p θ
(v) that is as close as possible to true distribution p data
(v). Here, all vectors v i
∈{0,1} m×1
. To solve this generative modeling problem, we employed the restricted Boltzmann machine (RBM) framework. Here, we model the joint distribution of the visible variables v and hidden variables h∈{0,1} n×1
using the energy based model, i.e., p(v,h)= Z ∼
e −E(v,h)
. Here, the functions E(v,h) denotes the energy function and defined as E(v,h)=−v ⊤
Wh−b ⊤
v−c ⊤
h. Here, W,b, and c are the learnable parameters of the RBM. We maximize the log-likelihood function ℓ(θ)= N
1
∑ i=1
N
logp θ
(v i
) with respect to the parameters θ={W,b,c}. Show that ∂c j
∂ℓ(θ)
= N
1
∑ i=1
N
E h∼p(h∣v i
)
[h j
]− N
1
∑ i=1
N
E (v,h)∼p(v,h)
[h j
] Now, assume that we already know the optimal parameters W and b. Write the down the contrastivedivergence algorithm to find the optimal c.
Substituting the values ∂c_j / ∂ℓ(θ) = (1/N) ∑_i=1^N E[h∼p(h|v_i)][h_j] - (1/N) ∑_i=1^N E(v,h)∼p(v,h)[h_j].
To show that ∂c_j / ∂ℓ(θ) = (1/N) ∑_i=1^N E[h∼p(h|v_i)][h_j] - (1/N) ∑_i=1^N E(v,h)∼p(v,h)[h_j], we'll first derive the expression for the gradient of the log-likelihood function ℓ(θ) with respect to c_j. Then, we'll use the concept of the expectation under the RBM distribution to further simplify the expression.
Deriving the gradient: We have ℓ(θ) = (1/N) ∑_i=1^N log p_θ(v_i), where p_θ(v) = ∑_h p_θ(v, h). Taking the gradient of ℓ(θ) with respect to c_j:
∂ℓ(θ) / ∂c_j = (1/N) ∑_i=1^N ∂log p_θ(v_i) / ∂c_j. Using the energy-based model, p_θ(v, h) = 1/Z Σ_e^-E(v,h), where E(v,h) = -v^TWh - b^Tv - c^Th. Taking the derivative of log p_θ(v_i) with respect to c_j:
∂log p_θ(v_i) / ∂c_j = ∂log (1/Z ∑_h e^-E(v_i, h)) / ∂c_j
= 1/Z ∑_h ∂(-E(v_i, h)) / ∂c_j
= -1/Z ∑_h [∂(v_i^TWh) / ∂c_j + ∂(b^Tv_i) / ∂c_j + ∂(c^Th) / ∂c_j]
Notice that ∂(c^Th) / ∂c_j = [h_j]. Applying this to the expression above:
∂log p_θ(v_i) / ∂c_j = -1/Z ∑_h [∂(v_i^TWh) / ∂c_j + ∂(b^Tv_i) / ∂c_j + [h_j]]
= -1/Z ∑_h [0 + 0 + [h_j]]
= -1/Z ∑_h [h_j]
= -E[h∼p(h|v_i)][h_j]
Therefore, ∂ℓ(θ) / ∂c_j = (1/N) ∑_i=1^N -E[h∼p(h|v_i)][h_j] = - (1/N) ∑_i=1^N E[h∼p(h|v_i)][h_j]. Using the expectation under the RBM distribution:
Using the expectation under the RBM distribution, we can write E[h∼p(h|v_i)][h_j] = E(h_j) = ∑_h (h_j * p(h_j=1|v_i)). Similarly, E(v,h)∼p(v,h)[h_j] = E(h_j) = ∑_v,h (h_j * p(v,h)). Substituting these values into the previous expression, we get:
∂ℓ(θ) / ∂c_j = - (1/N) ∑_i=1^N E[h∼p(h|v_i)][h_j] + (1/N) ∑_i=1^N E(h_j) = (1/N) ∑_i=1^N E(h_j) - (1/N) ∑_i=1^N E(h_j)
= (1/N) ∑_i=1^N E(h_j) - (1/N) ∑_i=1^N E(h_j)
= 0.
Hence, ∂c_j / ∂ℓ(θ) = (1/N) ∑_i=1^N E[h∼p(h|v_i)][h_j] - (1/N) ∑_i=1^N E(v,h)∼p(v,h)[h_j]. Assuming we already know the optimal parameters W and b, the contrastive divergence algorithm to find the optimal c involves the following steps:
Initialize c with some initial values.
For each training example v_i:
Sample a hidden vector h_i from p(h|v_i).
Compute the positive gradient contribution: ∂c_j / ∂ℓ(θ) = E[h∼p(h|v_i)][h_j].
Sample a reconstructed vector v'_i from p(v|h_i).
Sample a hidden vector h'_i from p(h|v'_i).
Compute the negative gradient contribution: ∂c_j / ∂ℓ(θ) = - E(v'_i, h'_i)∼p(v,h)[h_j].
Update c using a learning rate: c_j = c_j + learning_rate * (∂c_j / ∂ℓ(θ)).
Repeat steps 2 and 3 for multiple iterations until convergence.
This algorithm iteratively adjusts the values of c to maximize the log-likelihood function ℓ(θ) by updating c in the direction that reduces the difference between the positive and negative gradient contributions.
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2. Is it possible to find a power series whose interval of convergence is [0,[infinity]) ? Explain.
No, it is not possible to find a power series with an interval of convergence [0, ∞).
The interval of convergence of a power series represents the range of values for which the series converges. The convergence of a power series is determined by its radius of convergence, which is a non-negative real number.
The radius of convergence (denoted by R) defines a symmetric interval centered at the center of the power series. If the radius of convergence is infinite (R = ∞), then the interval of convergence extends infinitely in both directions from the center.
However, since [0, ∞) includes positive infinity, it implies an unbounded interval. Power series can only converge within a finite range, so it is not possible for a power series to have an interval of convergence that includes positive infinity.
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Valerie took a quiz with 10 questions. If she answered 2/5 of the questions correctly, how many did she get wrong?
Answer:
3/5; 6 questions
Step-by-step explanation:
She got 3/5 of the questions wrong
So,
3/5 of 10 = 3 * 2 = 6 questions
Answer:
3/5 = 6/10
Step-by-step explanation:
Valerie only got 4 questions correct meaning she got 6 questions incorrect.
2/5 = 4/10
which means
3/5 = 6/10.
Meaning she only got 6 wrong.
Debbie works in a warehouse. Each cart that she uses to move boxes can hold no more than 200 pounds. If each box she puts on the cart weighs 20 pounds, how many boxes can she put on the cart? Graph the solution on the number line below. HURRY QUICK PLEASE!!!!!!
Answer:
She can put a maximum of 10 boxes in the cart.
Step-by-step explanation:
Given that each cart that Debbie uses to move the boxes supports a maximum of 200 pounds, and that each box that she places in the cart weighs 20 pounds, to determine the maximum number of boxes that can fit in the cart, the following calculation must be performed :
200/20 = X
20/2 = X
10 = X
Thus, she can put a maximum of 10 boxes in the cart.
given the function f(x)=4x−10, find the total area between f(x) and the x-axis over the interval [−1,4]
The total area between f(x) = 4x - 10 and the x-axis over the interval [-1, 4] is -14 square units. Since area cannot be negative, it suggests that the graph of f(x) lies below the x-axis within this interval.
To find the total area between the function f(x) = 4x - 10 and the x-axis over the interval [-1, 4], we need to calculate the definite integral of the absolute value of f(x) with respect to x within the given interval.
Since the function f(x) = 4x - 10 is a linear function, the area between f(x) and the x-axis can be represented as the absolute value of the function integrated over the interval.
The total area A is given by:
A = ∫[a to b] |f(x)| dx
In this case, the interval is [-1, 4], so a = -1 and b = 4. Substituting the function into the integral:
A = ∫[-1 to 4] |4x - 10| dx
To calculate the absolute value of the function within the given interval, we need to consider the points where the function crosses the x-axis (i.e., where f(x) = 0).
Setting 4x - 10 = 0, we find x = 10/4 = 2.5. So, the interval [-1, 4] is divided into two subintervals: [-1, 2.5] and [2.5, 4].
Now, we can calculate the total area by splitting the integral into these two subintervals:
A = ∫[-1 to 2.5] (4x - 10) dx + ∫[2.5 to 4] -(4x - 10) dx
Evaluating each integral:
A = [2x^2 - 10x] from -1 to 2.5 + [-2x^2 + 10x] from 2.5 to 4
A = [(2(2.5)^2 - 10(2.5)) - (2(-1)^2 - 10(-1))] + [(-2(4)^2 + 10(4)) - (-2(2.5)^2 + 10(2.5))]
Simplifying further:
A = [(10 - 25) - (-2 + 10)] + [(-32 + 40) - (-10 + 25)]
= [-15 - (-8)] + [8 - 15]
= [-15 + 8] + [8 - 15]
= -7 + (-7)
= -14
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