The solution y = -1/(t^2 + 2t + 3) is defined for all real values of t, and the largest interval in which the solution is defined is (-∞, ∞).
To solve the initial value problem dy/dt = 2(t + 1)y^2, y(0) = -1/3, we can separate the variables and integrate both sides with respect to t.
Starting with the given differential equation:
dy/y^2 = 2(t + 1) dt
Integrating both sides:
∫(dy/y^2) = ∫(2(t + 1) dt)
Integrating the left side using the power rule for integration gives:
-1/y = t^2 + 2t + C1
To find the constant of integration, we use the initial condition y(0) = -1/3:
-1/(-1/3) = 0^2 + 2(0) + C1
3 = C1
Therefore, the equation becomes:
-1/y = t^2 + 2t + 3
Next, we can solve for y:
y = -1/(t^2 + 2t + 3)
Now, let's determine the largest interval in which the solution is defined. The denominator of y is t^2 + 2t + 3, which represents a quadratic polynomial. To find the interval where the denominator is non-zero, we need to consider the discriminant of the quadratic equation.
The discriminant, Δ, is given by Δ = b^2 - 4ac, where a = 1, b = 2, and c = 3. Substituting the values, we have:
Δ = (2)^2 - 4(1)(3) = 4 - 12 = -8
Since the discriminant is negative, Δ < 0, the quadratic equation t^2 + 2t + 3 = 0 has no real solutions. Therefore, the denominator t^2 + 2t + 3 is always positive and non-zero.
Hence, the solution y = -1/(t^2 + 2t + 3) is defined for all real values of t, and the largest interval in which the solution is defined is (-∞, ∞).
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PLEASE HELP!!!! Jul is using the graph of an exponential function to represent the value of an investment where x is the number of years Jul has owned the investment. Which statement correctly describes a key feature of the function?
Answer: the answer is D
Step-by-step explanation:
Trust me
the domain of the function is x greater than equal to 0.
Answer: The domain of the function is x>0. So D just took the test, got it right!
Step-by-step explanation:
River C is 400 miles longer than River D. If the sum of their lengths is 5560 miles, what is the length of each river?
Answer:
River D - 2580 miles
River C - 2980 miles
Step-by-step explanation:
Let the length of River D be represented with x
length of river C = 400 + x
total length of both rivers = 5560
x + 400 + x = 5560
2x + 400 = 5560
collect like terms
2x = 5560 - 400
2x = 5160
divide both sides of the equation by 2
x = 5160 / 2
x = 2580
length of river c = 400 + 2580 = 2980
What is the greatest common factor of 42 and 50
The greatest common factor of 42 and 50 is 2.
The given numbers are 42 and 50
The greatest common factor (GCF) of two numbers is the largest number that divides both of them evenly.
To find the GCF of 42 and 50,
We can start by listing the factors of each number.
The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.
Similarly, the factors of 50 are 1, 2, 5, 10, 25, and 50.
By comparing the factors,
We can see that the highest common factor is 2.
Therefore, the GCF of 42 and 50 is 2.
This means that 2 is the largest number that can divide both 42 and 50 without leaving a remainder.
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Which of the following is one solution to the expression (ax + b)(cx - d) = 0 ?
A. −b
B. d
C. −b/a
D. −d/c
Jim's Co. has set a requirement on stock items of a turnover ratio of 2.6 per year. It is examining three stocked items, A, B and C, which have to be bought in large amounts. As a result of the purchasing requirements, the maximum stock for A is $1,000, for B $1,200 and for C $2,500. If the average stock is assumed to be one-half the maximum stock, what would be the required annual sales of each of these items?
The required annual sales for stocked items A, B, and C would be $1,300, $1,560, and $3,250, respectively.
To calculate the required annual sales for each stocked item, we need to consider the turnover ratio and the maximum stock level. The turnover ratio indicates how many times the stock is sold and replaced within a year.
Given that the turnover ratio requirement is 2.6 per year, we can calculate the required annual sales for each item by multiplying the turnover ratio with the maximum stock level.
For item A, the maximum stock level is $1,000, and the required annual sales would be 2.6 times $1,000, which equals $2,600.
Similarly, for item B, the maximum stock level is $1,200, and the required annual sales would be 2.6 times $1,200, which equals $3,120.
For item C, with a maximum stock level of $2,500, the required annual sales would be 2.6 times $2,500, which equals $6,500.
However, since the average stock is assumed to be one-half the maximum stock, we need to adjust the required annual sales accordingly. The average stock for each item would be $500 for A, $600 for B, and $1,250 for C. Therefore, the required annual sales for A would be $2,600 minus $500, which equals $1,300. For B, it would be $3,120 minus $600, which equals $1,560. And for C, it would be $6,500 minus $1,250, which equals $3,250.
In summary, the required annual sales for items A, B, and C would be $1,300, $1,560, and $3,250, respectively.
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You are interested in estimating the thic THCurwis of the local adult population of female white-tailed deer (doe). From past data, you estimate that the standard deviation of all adult female white-tailed deer in this region to be 18 pounds. What sample size would you need to in order to estimate the mean weight of all female white-tailed deer, with a 90% confidence level, to within 5 pounds of the actual weight?
The sample size required to estimate the mean weight of all female white-tailed deer with a 90% confidence level within 5 pounds of the actual weight is 43.
In order to estimate the sample size required to estimate the mean weight of all female white-tailed deer with a 90% confidence level within 5 pounds of the actual weight, the following steps are to be followed:
Step 1: Determine the confidence level and the maximum allowable error
The given confidence level is 90%.
The maximum allowable error is 5 pounds.
Step 2: Determine the population standard deviationThe population standard deviation is given as σ = 18 pounds.
Step 3: Determine the critical value
The critical value corresponding to a 90% confidence level is 1.645.
Step 4: Calculate the sample size formula to calculate the sample size is given as
n = [(z-value)² × σ²] / E² Where n = sample size
z-value = critical value
σ = population standard deviation = maximum allowable error
Substituting the given values in the above formula, we get,n = [(1.645)² × (18)²] / (5)²= 42.68≈43 (approx)
Therefore, the sample size required to estimate the mean weight of all female white-tailed deer with a 90% confidence level within 5 pounds of the actual weight is 43. Answer: 43
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Find the surface area of the prism. Whoever solves this will guarantee get Brainliest Answer!!!
To find the total surface area of a prism, you need to calculate the area of two polygonal bases, i.e., the top face and bottom face. And then calculate the area of lateral faces connecting the bases. Add up the area of the two bases and the area of the lateral faces to get the total surface area of a prism.
Hope this helps!
Find the axis of symmetry and the vertex of the graph (Desmos)
Answer:
The axis of symmetry is x=3, the vertex of the graph is (3,-2)
Step-by-step explanation:
[tex]f(x)=(x^{2} -6x+9)-9+7[/tex]
[tex]=(x-3)^{2} -2[/tex]
Test for a significant change in the attitude toward increased federal funding for stem cell research, as measured on an ordinal scale survey, before and after 22 people hear a discussion of the issue on a network news program.
The steps to test for a significant change in the attitude toward increased federal funding for stem cell research.
As measured on an ordinal scale survey, before and after 22 people hear a discussion of the issue on a network news program are as follows:
Step 1: State the null and alternative hypotheses
In this case, the null hypothesis states that there is no significant change in attitude towards increased federal funding for stem cell research before and after 22 people hear a discussion of the issue on a network news program. The alternative hypothesis states that there is a significant change in attitude towards increased federal funding for stem cell research before and after 22 people hear a discussion of the issue on a network news program.
Step 2: Set the level of significance
The level of significance, denoted by alpha, is the probability of rejecting the null hypothesis when it is true. A common level of significance is 0.05. This means that there is a 5% chance of rejecting the null hypothesis when it is true.
Step 3: Calculate the test statistic
To calculate the test statistic, we can use the Wilcoxon signed-rank test. This test compares the scores before and after the treatment and calculates the difference between them. The Wilcoxon signed-rank test is used for paired samples or repeated measures. It is a nonparametric test and is used when the data is not normally distributed or when the data is ordinal.
Step 4: Determine the critical value
To determine the critical value, we need to look up the value in the Wilcoxon signed-rank table. The critical value is the smallest value that would lead to the rejection of the null hypothesis.
Step 5: Compare the test statistic to the critical value
If the test statistic is greater than the critical value, we reject the null hypothesis. If the test statistic is less than the critical value, we fail to reject the null hypothesis.
Step 6: Interpret the results
If we reject the null hypothesis, we can conclude that there is a significant change in attitude towards increased federal funding for stem cell research before and after 22 people hear a discussion of the issue on a network news program. If we fail to reject the null hypothesis, we cannot conclude that there is a significant change in attitude towards increased federal funding for stem cell research before and after 22 people hear a discussion of the issue on a network news program.
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The ordinal scale survey is the type of scale utilized for the measurement of attitude, and it is used to rank the participant's responses on an order (ordinal) from smallest to greatest.
The survey measured the attitude toward increased federal funding for stem cell research, and it was conducted on 22 people before and after a discussion on the issue on a network news program.
A statistical analysis of the survey responses can be conducted using the Wilcoxon Signed Rank Test to test for a significant change in attitude.
The Wilcoxon Signed Rank Test is an ideal nonparametric test that tests for changes in attitudes before and after a discussion on the issue on a network news program.
The Wilcoxon Signed Rank Test compares the scores of each participant's responses on the ordinal scale before and after the discussion.
After computing the differences between the scores, the test ranks the differences, and then the sum of the positive and negative ranks is calculated.
The calculated sum is then compared with the Wilcoxon Signed Rank Test critical values, which depend on the number of pairs tested, which in this case, is 22.
If the calculated sum is higher than the critical value, then there is a significant change in attitude towards increased federal funding for stem cell research after the discussion on the network news program.
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The jaguar is a top predator that helps to regulate other population in the rainforest. It produces waste that are broken down to nutrients by decomposers. Microorganisms live in its fur and it may also be home to some parasites. This description describes the jaguar's __________ ?
A. Habitat
B. Niche
C. Awesome ninja-like skills
D. Trophic Level
Answer: B. Niche
Explanation:
Definition of niche: a niche is the match of a species to a specific environmental condition. It describes how an organism or population responds to the distribution of resources and competitors and how it in turn alters those same factors.
Answer: Niche
Niche is basically like a type of place of something or someone
11. Which set of ordered pairs represents y as a function of x?
A. {(-4,-3), (-4,-2), (-3,-3), (-3,-2)}
B. {(2.0), (4,0), (4,2), (6, 2)}
C. {(6,-2), (6.0), (6,2), (6,4)}
D. {(0, 0), (2, -4), (4, -8), (6,-12)}
Answer is C. {(6,-2), (6.0), (6,2), (6,4)}
Step-by-step explanation:
Your welcome
ill put this up for 50 just please someone help me.
Answer:
1) Let the dimensions be x for greater size and 1 for unit size
Product is:
x² + 3x + 2x + 6 = x² + 5x + 6Factors are:
(x + 3) and (x + 2)2)
Product is:
2x² + 3x + 2x + 3 = 2x² + 5x + 3Factors are:
(2x + 3) and (x + 1)Let T be a linear transformation from P2 to R², defined by T(p) = [p(0) p(1)] polynomial p(t) in P2. Find bases for kernel of T and range of T.
For the linear transformation we have:
Basis for the kernel of T (Ker(T)): {[t² - t]}
Basis for the range of T (Range(T)): {[1, 1], [1, 0], [0, 1]}
How to find the bases for kernel of T and range of T?To find the bases for the kernel and range of the linear transformation T: P2 -> R², defined by T(p) = [p(0), p(1)], we need to understand the properties of T and solve for the appropriate vectors.
Kernel of T (Nullspace):
The kernel of T, denoted as Ker(T), consists of all polynomials p in P2 such that T(p) = [p(0), p(1)] = [0, 0]. In other words, the kernel contains all polynomials that get mapped to the zero vector in R².
Let's solve for the kernel by setting up the system of equations:
p(0) = 0
p(1) = 0
Since we are dealing with polynomials of degree at most 2, let's consider a general polynomial p(t) = at² + bt + c.
Substituting into the equations:
(0)a + (0)b + c = 0 -> c = 0
(1)a + (1)b + c = 0 -> a + b = 0 -> b = -a
Thus, any polynomial of the form p(t) = at² - at, where a is a scalar, will be in the kernel of T. Therefore, a basis for the kernel is [t² - t].
Range of T:
The range of T, denoted as Range(T), consists of all vectors in R² that can be obtained by applying the linear transformation T to some polynomial in P2.
To find the range, we need to determine all possible outputs of T(p) for polynomials p in P2.
Let's consider a general polynomial p(t) = at² + bt + c and apply T(p):
T(p) = [p(0), p(1)] = [a(0)² + b(0) + c, a(1)² + b(1) + c] = [c, a + b + c]
So, any vector [x, y] in the range of T must satisfy x = c and y = a + b + c for some a, b, c.
In R², any vector [x, y] can be written as [x, y] = [c, a + b + c] = c[1, 1] + a[1, 0] + b[0, 1], where a, b, c are scalars.
So, the basis for the range of T is {[1, 1], [1, 0], [0, 1]}.
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STRESSING‼️ PLEASE HELP❗️❗️
plssss help
use the property that it says to solve the proportion ty
Answer:
the answer is 45 over 6
Step-by-step explanation:
A shortcut is "2 x _____ is 6", which is 3.
Multiply 15 by 3 also, to get 45,
45/6
A car dealership offers two types of discounts.
• Discount 1: Take 5% off the original price of a car built last year and then receive a
$3,500 rebate.
• Discount 2: Take 10% off the original price of a car built this year and then receive a
$1,250 rebate.
A customer is deciding between two cars.
• Car R was built last year and has an original price of $25,340.
• Car S was built this year and has an original price of $22,860.
Based on this information, which statement is true?
A The customer would pay $19,324 for Car S.
B The customer would pay $24,073 for Car R.
C The customer would pay $21,824 for Car S.
D The customer would pay $23,107 for Car R.
Answer:
The customer would pay $19,324 for Car S.
Step-by-step explanation:
Take Car S
$22,860 x .1 (ten percent) = $2,286
$22,860 (original car price) - $2,286 (minus the 10%) = $20,574 (new car price)
$20,574 - rebate of $1,250 = $19,324
The customer would pay $19,324 for Car S.
All other do not come out but this one does, so that is the only option for this!
IM GIVING BRAINLIEST!!PLEASE HELP!!
Answer:
D
Step-by-step explanation:
(x - 3)(x + 4) = x^2 + x - 12
Calculate the value of (6.9x10^-3)x(2x10^9) Give your answer in standard form.
Question 1[16 marks] Consider the following optimisation problem max f(x, y) = t √ x y, subject to tx^2 + y ≤ 5, x ≥ 0, y ≥ 0.
a) Solve the problem for t = 1.
b) State and explain the content of the envelope theorem.
c) What is the marginal effect on the solution if the constant t is increased?
(a) Optimization problem: The optimization problem is shown below:
max f(x, y) = t √ x y, subject to tx² + y ≤ 5x ≥ 0y ≥ 0
Solving the problem for t = 1,t = 1f(x,y) = √xytx² + y ≤ 5x ≥ 0y ≥ 0.
The Lagrangian function for this problem is:
L(x, y, λ) = t √ xy + λ(5 - tx² - y)
We set the partial derivative of L with respect to x to zero:
∂L/∂x = t(0.5√y)/√x + (-2λtx) = 0
We then obtain:
(1) 0.5t√y/√x = 2λtxIf we set the partial derivative of L with respect to y to zero, we obtain:
(2) 0.5t√x/√y + λ(-1) = 0
Multiplying both sides by (-1), we obtain:
(3) -0.5t√x/√y = λ We set the partial derivative of L with respect to λ to zero, we obtain:
(4) 5 - tx² - y = 0Substituting Equation (3) into Equation (1), we obtain:
(5) 0.5t√y/√x = -2(5 - tx² - y)x
Substituting Equation (5) into Equation (4), we obtain:
(6) 5 - tx² - 2x²(5 - tx² - y)² = 0
After expanding Equation (6), we obtain a fourth-order equation in y. Solving this equation leads to:(7) y = 5 - tx²/3
We then substitute y into Equation (3) to obtain: x = 5/2t²From Equation (7), we obtain: y = 5 - tx²/3=5-5/3*2.5=2.7778
(b) Envelope theorem
According to the Envelope Theorem, the marginal effect of a parameter on an optimal solution is equal to the partial derivative of the optimal value with respect to that parameter. This means that if a parameter changes slightly, the change in the optimal value can be estimated using the first-order approximation. (c) Increasing the constant tIf we increase the constant t, the optimal x and y will also increase. This is because an increase in t will lead to a higher value of f(x, y).
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Darius wants to buy a new car. The car he chooses has a total purchase price of $18,500. Darius uses a multi-offer website to apply for a car loan. He receives three offers with minimum payments he can afford. The terms for each loan are shown in the table.
Darius belive he should chose loan option B. Which of the following statements about Darius’s choice is true?
A. Darius should choose loan option B because it has the shortest loan term.
B. Darius should not choose loan option B because he will pay $398.52 more in interest than on loan option C.
C. Darius should choose loan option B because he will pay $280.44 less in interest than on loan option A.
D. Darius should not choose loan option B because it does not have the lowest interest rate.
Answer:
the correct answer is B, Darius should not choose loan option B because he will pay $398.52 more in interest than on loan option C.
Step-by-step explanation:
I also had this question, and it showed that I got it correct :)
Please help meee! I will give brainleiest!
Answer:
64? I'm not sure but because if 1= girls (8 girls) and boys is 8 times more you do 8x8 I think sorry if I'm wrong
Answer:
There are 64 boys
Step-by-step explanation:
If there are 8 girls it would be 8*1
so you would have to multiply 8 by the ratio of boys which is 8*8
which equals 64 boys to 8 girls (8:1)
Your welcome
A company is designing a new cylindrical water bottle the volume of the bottle by 204 cm the height of the water bottles 8.9 cm3 what is the radius of a water bottle use 3.14 for pie
Answer:
2.70cm
Step-by-step explanation:
Given data
Answer:
2cm
Step-by-step explanation:
Given data
Capacity/volume= 204 cm
Height= 8.9 cm
Required
The radius r
let us apply the expression for the volume of a cylinder
V=πr^2h
204=πr^2*8.9
204=3.14r^2*8.9
r^2= 204/27.946
r^2=7.30
r= √7.30
r=2.70cm
Hence the radius is 2.70cm
mark+is+shopping+during+a+computer+store’s+20%+sale.+he+is+considering+buying+computers+that+range+in+cost+from+$500+to+$1000.+how+much+is+the+least+expensive+computer+after+the+20%+discount?
The least expensive computer after the 20% discount would be $400.
To calculate the price of the least expensive computer after the 20% discount, we need to find 20% of the original price and subtract it from the original price.
Let's assume the original price of the least expensive computer is x. The discount of 20% can be calculated as 0.20 * x. To find the discounted price, we subtract the discount from the original price: x - 0.20 * x = 0.80 * x.
Since we know that the cost of the least expensive computer ranges from $500 to $1000, we can substitute x with $500 and calculate the discounted price: 0.80 * $500 = $400. Therefore, the least expensive computer after the 20% discount would be $400.
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Let Z = {] ² | c=b}. ER}. Prove that Z is a subspace of R2x2. for some beR Prove that Y is not a subspace of R2×2,
To prove that Z = {[b² | c=b] | b, c ∈ ℝ} is a subspace of ℝ²x², we need to show that Z satisfies the three properties of a subspace.
To prove that Y = {A ∈ ℝ²x² | A is an upper triangular matrix} is not a subspace of ℝ²x², we only need to show that it fails to satisfy one of the three properties.
For Z to be a subspace of ℝ²x², it needs to satisfy closure under addition, closure under scalar multiplication, and contain the zero vector.
1. Closure under addition: Let A = [b₁² | c₁=b₁] and B = [b₂² | c₂=b₂] be two matrices in Z. Their sum, A + B, is [b₁² + b₂² | c₁ + c₂ = b₁ + b₂]. Since b₁ + b₂ is a real number, A + B is also in Z. Hence, Z is closed under addition.
2. Closure under scalar multiplication: Let A = [b² | c=b] be a matrix in Z, and k be a scalar. The scalar multiple kA is [k(b²) | k(c) = kb]. Since kb is a real number, kA is also in Z. Therefore, Z is closed under scalar multiplication.
3. Contains the zero vector: The zero vector in ℝ²x² is the matrix [0 0 | 0 = 0]. This matrix satisfies the condition c = b, so it is in Z.
Thus, Z satisfies all the properties and is a subspace of ℝ²x².
For Y to be a subspace of ℝ²x², it needs to satisfy the three properties mentioned earlier. However, Y fails to satisfy closure under addition since the sum of two upper triangular matrices may not always be an upper triangular matrix. Hence, Y is not a subspace of ℝ²x².
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What is the solution to the system of equations graphed below?
a. (2,-3)
b. (-3,2)
c. (-2,3)
d. (3,-2
plz help i’m in a timed test
The equation for a parabola has the form y=az? + bz +c, where a, b, and care constants and a 70. Find an equation for the parabola that passes through the points (-1,0), (1,6), and (-3,-14). Answer: y = __
The equation for the parabola that passes through the points (-1,0), (1,6), and (-3,-14) is y = -2x² + 3x + 1.
To find the equation of a parabola, we need to substitute the given points into the general form of the equation, y = az² + bz + c, and solve for the unknown constants a, b, and c. In this case, we have three points: (-1,0), (1,6), and (-3,-14). We can plug in the x and y values of each point into the equation to obtain three equations.
For the point (-1,0):
0 = a(-1)² + b(-1) + c
For the point (1,6):
6 = a(1)² + b(1) + c
For the point (-3,-14):
-14 = a(-3)² + b(-3) + c
Now we have a system of three equations with three unknowns (a, b, and c). Solving this system will give us the values of a, b, and c, which we can then substitute back into the general equation to obtain the specific equation of the parabola.
To solve the system, we can use various methods such as substitution, elimination, or matrices. However, to keep the explanation concise, I'll provide the solution directly.
Solving the system of equations, we find:
a = -2b = 3c = 1Substituting these values back into the general equation, we get:
y = -2x² + 3x + 1
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Please help I'm begging a ACE OR GENES To help me please please help please please ASAP please please help please please ASAP please please help
Answer:
3/2
Step-by-step explanation:
WX/AB = 12/8 = 3/2
Answer: 3/2
The covariance of random variables X, Y is defined as Cov(X,Y) = E[(X – Ux)(Y – My)] where Úx = E(X) and My = E(Y). Note: Var(X) = Cov(X,X).
(d) Show that [E(XY)]? < E(X)E(Y). Hint: Let Z=X+ay,
We have shown that [E(XY)]^2 < E(X)E(Y), as required.
To show that [E(XY)]^2 < E(X)E(Y), we can follow the hint provided and introduce a new random variable Z = X + aY, where 'a' is a constant.
First, let's expand the expression E(XY) using the law of iterated expectations:
E(XY) = E[E(XY|Z)]
Now, substituting Z = X + aY into the conditional expectation:
E(XY) = E[E(X(X + aY)|Z)]
= E[E(X^2 + aXY|Z)]
Expanding the inner expectation:
E(XY) = E[X^2 + aXY]
Next, let's square both sides of the inequality to be proved:
[E(XY)]^2 < E(X)E(Y)
(E[X^2 + aXY])^2 < E(X)^2E(Y)^2
Expanding the square:
E(X^2)^2 + 2aE(X^2)E(XY) + a^2E(XY)^2 < E(X)^2E(Y)^2
Since E(X^2) is the variance of X (Var(X)), we can rewrite it as:
Var(X) + [E(X)]^2
Using the covariance formula, Cov(X,Y) = E[(X - Ux)(Y - My)], we can rewrite the second term as:
Cov(X,Y) + [E(X)][E(Y)]
Substituting these expressions back into the inequality, we have:
Var(X) + [E(X)]^2 + 2a(Cov(X,Y) + [E(X)][E(Y)]) + a^2[E(XY)]^2 < E(X)^2E(Y)^2
Simplifying the equation, we have:
Var(X) + 2aCov(X,Y) + a^2[E(XY)]^2 < 0
This inequality holds true since the left-hand side of the equation is a quadratic expression in 'a' and the coefficient of the quadratic term is positive (Var(X)). Since the inequality holds for all values of 'a', it must hold when 'a' is zero. Therefore, we have:
Var(X) + 0 + 0 < 0
Which is not possible, thus proving that [E(XY)]^2 < E(X)E(Y).
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Solve the equation −6 +x/4 = −5
Pls help
Answer:
x=4
Step-by-step explanation:
Answer:
x = 4
Step-by-step explanation:
-6 + x/4 = -5
-6+6 + x/4 = -5+6
x/4 = 1
x/4 *4 = 1*4
x = 4
hope this helped :)
A summer camp is organizing a hike and needs to buy granola bars for the campers. The granola bars come in small boxes and large boxes. Each small box has 6 granola bars and each large box has 24 granola bars. The camp bought 4 times as many small boxes as large boxes, which altogether had 96 granola bars. Graphically solve a system of equations in order to determine the number of small boxes purchased, x,x, and the number of large boxes purchased, yy.
Answer:
Let's define the variables:
x = number of small boxes bought.
y = number of large boxes bought.
Then the total number of granola bars is:
x*6 + y*24
We also know that "The camp bought 4 times as many small boxes as large boxes"
Then:
x = 4*y
and "...which altogether had 96 granola bars."
The total number of granola bars is 96, then:
x*6 + y*24 = 96
Then the system of equations is:
x = 4*y
x*6 + y*24 = 96
We want to solve this graphically.
Then we first need to isolate the same variable in both equations.
We can see that in the first one x is already isolated, so let's isolate x in the second equation:
x*6 = 96 - y*24
x = (96 - y*24)/6
x = 16 - y*4
Now we have the equations:
x = 4*y
x = 16 - y*4
To solve this graphically we need to graph both fo these lines and see in which point the lines do intersect.
The point where the lines intersect is the solution of the system.
The graph can be seen below.
We can see that the lines do intersect at the point (2, 8)
This means that the camp bought 2 large boxes and 8 small boxes.