Solve for the value of a
Answer:
a=7
Step-by-step explanation:
These two angles are complementary and the sum of their measures is 90°.
Therefore, we can create the equation: (5a+3)+52=90.
1. combine like terms. the equation becomes 5a+55=90.
2. -55 to both sides of the equation. the equation becomes 5a=35.
3. /5 to both sides of the equation. we can reach the conclusion that a=7
find a parametric representation for the surface.
part of the surface of the sphere x² + y² + z² = 4 that lies above the cone z = √x²+y².
The parametric representation for the surface is x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), z = ρcos(φ) with the restrictions 0 ≤ ρ ≤ 2, 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π/4.
To find a parametric representation for the surface that lies above the cone z = √(x² + y²) and is part of the sphere x² + y² + z² = 4, we can express the surface in terms of spherical coordinates.
In spherical coordinates, the sphere x² + y² + z² = 4 can be represented as:
ρ² = 4
ρ = 2
Since we want to consider only the part of the sphere above the cone, we restrict the values of ρ to be between 0 and 2.
The cone z = √(x² + y²) in spherical coordinates is expressed as:
z = ρcos(φ)
Combining these equations, we can find the parametric representation for the desired surface:
x = ρsin(φ)cos(θ)
y = ρsin(φ)sin(θ)
z = ρcos(φ)
However, we need to restrict the values of ρ and φ to only the part of the surface above the cone. This means that ρ should range from 0 to 2, and φ should range from 0 to the angle that corresponds to the cone z = √(x² + y²).
Let's find the range of φ by substituting the equation for the cone into the equation for z:
z = ρcos(φ)
√(x² + y²) = ρcos(φ)
Since x² + y² = ρ²sin²(φ) (using the spherical coordinate expressions for x and y), we can rewrite the equation as:
√(ρ²sin²(φ)) = ρcos(φ)
ρsin(φ) = ρcos(φ)
tan(φ) = 1
Solving for φ, we find φ = π/4.
Therefore, the parametric representation for the surface is:
x = ρsin(φ)cos(θ)
y = ρsin(φ)sin(θ)
z = ρcos(φ)
with the restrictions:
0 ≤ ρ ≤ 2
0 ≤ θ ≤ 2π
0 ≤ φ ≤ π/4
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1. A random sample of 400 married couples was selected from a large population of married couples. There were 20 couples in which the wife was taller than her husband, and there were 380 couples in which the wife was shorter that her husband. Find a 95 percent confidence interval for the proportion of married couples in the population for which the wife is taller than her husband. Interpret your interval in the context of this question.
Answer:
[tex]CI = (0.028636,0.071364)[/tex]
I am 95% confident that the true proportion of couples where the wife is taller than her husband is captured in the interval (.028, .071)
Step-by-step explanation:
Given
[tex]n = 400[/tex]
[tex]x = 20[/tex] --- taller wife
[tex]y = 380[/tex] --- shorter wife
Required
Determine the 95% confidence interval of taller wives
First, calculate the proportion of taller wives
[tex]\hat p = \frac{x}{n}[/tex]
[tex]\hat p = \frac{20}{400}[/tex]
[tex]\hat p = 0.05[/tex]
The z value for 95% confidence interval is:
[tex]z = 1.96[/tex]
The confidence interval is calculated as:
[tex]CI = \hat p \± z \sqrt{\frac{\hat p (1 - \hat p)}{n}}[/tex]
[tex]CI = 0.05 \± 1.96* \sqrt{\frac{0.05 (1 - 0.05)}{400}}[/tex]
[tex]CI = 0.05 \± 1.96 * \sqrt{\frac{0.0475}{400}}[/tex]
[tex]CI = 0.05 \± 1.96 * \sqrt{0.00011875}[/tex]
[tex]CI = 0.05 \± 1.96 * 0.01090[/tex]
[tex]CI = 0.05 \± 0.021364[/tex]
This gives:
[tex]CI = (0.05 - 0.021364,0.05 + 0.021364)[/tex]
[tex]CI = (0.028636,0.071364)[/tex]
Rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.
X = 36y²
The given equation, X = 36y², represents a parabola. In standard form, the equation can be rewritten as y² = (1/36)x. The vertex (V) is located at the origin (0, 0), the focus (F) is at (0, 1/4), and the directrix (d) is the horizontal line y = -1/4.
To rewrite the equation X = 36y² in standard form, we divide both sides by 36 to get y² = (1/36)x. This form represents a parabola with its vertex at the origin (0, 0).
In standard form, the equation of a parabola can be written as y² = 4px, where p is the distance from the vertex to the focus and also the distance from the vertex to the directrix. In this case, p = 1/4.
Therefore, the vertex (V) is located at (0, 0), the focus (F) is at (0, 1/4), and the directrix (d) is the horizontal line y = -1/4.
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Find the total lateral area of the following
cone. Leave your answer in terms of a.
4 cm
3 cm
LA = ? cm2
Answer:
15π cm²
Step-by-step explanation:
The total lateral area of a cone
= πr√h² + r²
= √h² + r² = l
h = Height = 4cm
r = radius = 3cm
Hence:
= π × 3 √4² + 3²
= 3π × √16 + 9
= 3π × √25
= 3π × 5
= 15π cm²
Jake writes this word expression.
the product of 9 and m
Enter an algebraic expression for the word expression.
The expression is m.
Then, evaluate the expression for m = 6.
PLSS HELP IMMEDIATELY!!! i’ll give brainiest if u don’t leave a link!
Answer: Evaluate the findings to compare to his hypothesis
Step-by-step explanation: Since the biologist already has the findings and has a hypothesis, he now has to compare both of them together.
The table shows the amounts A (in billions of dollars) budgeted for national defense for the years 1998 to 2004.
Cierra is buying juice. She needs 5 liters. A half liter juice cost $2.86. A 250-milliliter container of juice costs $1.05. What should Cierra buy so she gets 5 liters at the lowest price?
Answer: 250 mL Juice container
Step-by-step explanation:
Given
Half liter juice costs $2.86 i.e.
[tex]\dfrac{1}{2}\ L\rightarrow\$2.86\\\\1\ L\rightarrow\dfrac{2.86}{\frac{1}{2}}=\$5.72\\\\5\ L\rightarrow\$28.6[/tex]
A 250 mL juice costs $1.05 i.e.
[tex]250\ mL=0.25\ L\rightarrow \$1.05\\\\1\ L\rightarrow \dfrac{1.05}{0.25}=\$4.2\\\\\Rightarrow 5\ L\rightarrow \$21[/tex]
The cost of 250 mL Juice packet is low for 5 L quantity, therefore, Cierra must buy 250 mL Juice container
The distribution of actual weights of 8-ounce chocolate bars produced by a certain machine is Normal with mean 8.1 ounces and standard deviation 0.1 ounces. Company managers do not want the weight of a chocolate bar to fall below 8 ounces, for fear that consumers will complain. (a) Find the probability that the weight of a randomly selected candy bar is less than 8 ounces Forty candy bars are selected at random and their mean weight is computed. (b) Calculate the mean and standard deviation of the sampling distribution of (c) Find the probability that the mean weight of the forty candy bars is less than 8 ounces. (d) Would your answers to (a), (b), or (c) be affected if the weights of chocolate bars produced by this machine were distinctly non-Normal? Explain.
a. the probability that the weight of a randomly selected candy bar is less than 8 ounces is approximately 0.1587 or 15.87%. b. the standard deviation of the sampling distribution is 0.1 / sqrt(40) ≈ 0.0159 ounces. c. the probability that the mean weight of the forty candy bars is less than 8 ounces is almost 0 or very close to 0.
(a) The probability that the weight of a randomly selected candy bar is less than 8 ounces can be found by calculating the cumulative probability using the Normal distribution. Given that the distribution of weights is Normal with a mean of 8.1 ounces and a standard deviation of 0.1 ounces, we want to find P(X < 8), where X represents the weight of a candy bar.
Using the properties of the Normal distribution, we can standardize the variable X using the formula Z = (X - μ) / σ, where Z is the standard normal random variable, μ is the mean, and σ is the standard deviation.
For our case, we have Z = (8 - 8.1) / 0.1 = -1.
Using a standard normal distribution table or a calculator, we find that the cumulative probability for Z = -1 is approximately 0.1587. Therefore, the probability that the weight of a randomly selected candy bar is less than 8 ounces is approximately 0.1587 or 15.87%.
(b) The mean of the sampling distribution of the sample mean can be calculated as the same as the mean of the population, which is 8.1 ounces.
The standard deviation of the sampling distribution of the sample mean, also known as the standard error of the mean, can be calculated using the formula σ / sqrt(n), where σ is the standard deviation of the population and n is the sample size.
In our case, the standard deviation of the population is 0.1 ounces, and the sample size is 40 candy bars. Therefore, the standard deviation of the sampling distribution is 0.1 / sqrt(40) ≈ 0.0159 ounces.
(c) To find the probability that the mean weight of the forty candy bars is less than 8 ounces, we can again use the properties of the Normal distribution. Since the mean and standard deviation of the sampling distribution are known, we can standardize the variable using the formula Z = (X - μ) / (σ / sqrt(n)).
In this case, we have Z = (8 - 8.1) / (0.0159) ≈ -6.29.
Using a standard normal distribution table or a calculator, we find that the cumulative probability for Z = -6.29 is extremely close to 0. Therefore, the probability that the mean weight of the forty candy bars is less than 8 ounces is almost 0 or very close to 0.
(d) The answers to (a), (b), and (c) would not be affected if the weights of chocolate bars produced by this machine were distinctly non-Normal. This is because of the Central Limit Theorem, which states that regardless of the shape of the population distribution, as the sample size increases, the sampling distribution of the sample mean approaches a Normal distribution.
In our case, we have a sufficiently large sample size of 40, which allows us to rely on the Central Limit Theorem. As long as the sample size is large enough, the sampling distribution of the sample mean will still be approximately Normal, even if the population distribution is non-Normal.
Therefore, we can still use the Normal distribution to calculate probabilities and determine the mean and standard deviation of the sampling distribution, regardless of the population distribution being non-Normal.
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Does someone mind helping me with this? I thought I had it but didnt. Thank you!
When x = 5, y = -1
When x = 6, y = 0
When x = 9, y = 1
When x = 14, y = 2
Suppose that $575.75 is invested in a savings account with an APR of 12% compounded monthly. What is the future value of the account in 5 years?
Answer:
FV= $1,045.96
Step-by-step explanation:
Giving the following information:
Initial investment (PV)= $575.75
Number of periods (n)= 15*5= 60 months
Interest rate (i)= 0.12 / 12= 0.01
To calculate the future value (FV), we need to use the following formula:
FV= PV*(1+i)^n
FV= 575.75*(1.01^60)
FV= $1,045.96
The initial size of a bacteria culture is 1000. After one hour the bacteria count is 8000. After how many hours will the bacteria population reach 15000? Assume the population grows exponentially.
Answer: Let’s assume that the bacteria population grows exponentially according to the formula P(t) = P0 * e^(kt), where P0 is the initial population, k is the growth rate, t is time in hours, and e is the mathematical constant approximately equal to 2.71828. We know that at time t = 0, the population is P(0) = 1000. After one hour, the population is P(1) = 8000. We can use this information to solve for the growth rate k. Substituting the values into the formula, we get: 8000 = 1000 * e^(k * 1) Dividing both sides by 1000, we get: 8 = e^k Taking the natural logarithm of both sides, we get: ln(8) = k Now that we have solved for k, we can use the formula to find out when the population will reach 15000. 15000 = 1000 * e^(ln(8) * t) Dividing both sides by 1000, we get: 15 = e^(ln(8) * t) Taking the natural logarithm of both sides, we get: ln(15) = ln(8) * t Dividing both sides by ln(8), we get: t = ln(15)/ln(8) ≈ 1.71 hours So it will take approximately 1.71 hours for the bacteria population to reach 15000. Received message.
Given the line y = 3x + 5:
Write the equation of a line, in point-slope form, that is parallel to the original and passes through the point (7,1)
match the lines l1 (blue), l2 ( red) and l3 (green) with the slopes by placing the letter of the slopes next to each set listed below:
I apologize, I cannot visually perceive or manipulate colors or lines directly. However, if you provide me with the equations of the lines or any additional information about their slopes, I would be more than happy to help you match the slopes with the corresponding lines.
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To match the lines (l1, l2, l3) with their corresponding slopes, we need to assign letters representing the slopes to each set.
Without specific information about the lines l1, l2, and l3, it is not possible to determine the exact slopes or their corresponding letters. In order to match the lines with their slopes, we would need additional details such as the equations of the lines or the coordinates of points on the lines.
The slope of a line represents the rate at which the line changes vertically (y-axis) with respect to the horizontal (x-axis). It is typically denoted by the letter "m" in mathematical notation.
To accurately match the lines l1, l2, and l3 with their respective slopes, we require more information about the lines themselves. Once the equations or coordinates are provided, we can calculate the slopes using the formula for slope, which is the change in y divided by the change in x.
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The charge to ship a package from one town to another, C, is given below as a function of the weight of the object, w, in pounds.
C = $3.50 + $0.55w
If the shipping cost for Cathy's item was $7.02, what was its weight?
A. 6.4 pounds
B. 64 pounds
C. 16.4 pounds
D. 0.64 of a pound
Answer:
Option A
Step-by-step explanation:
The total shipping cost given was $7.02.
C = 7.02
Solve for 'w'.
[tex]C = $3.50 + $0.55w\\\\7.02=3.50+0.55w\\\\3.52=0.55w\\\\\boxed{6.4=w}[/tex]
Hope this helps.
someone help this question is worth 50 points! What ratios are equivalent to the ratio 24:4
A.) 6:1
B.) 12:2
C.) 4:24
D.) 48:8
E.) 18:3
F:) 1:6
Step-by-step explanation:
just put the ratios into a fraction if x:y then x/y
A.)6/1=6
B.)12/2=6
C.)4/24=1/6
D.)48/8=6
E.)18/3=6
F.)1/6
24/4=6 so A, B, D, and E are equivilant to the ratio 24/4
Hope that helps :)
Answer:
6:1 ,12:2, 48:8
Step-by-step explanation:
24:4
24 ÷ 4 =6 and 4÷4 =1
6:1
24:4
24÷2 =12 , 4÷2=2,
12:2
48:8
24×2=48, 4×2=8
48:8
Hey Guys,.I just wanted to check. Is this correct? :V
Answer:
It's correct.
Step-by-step explanation:
- - - - - - - - - - - - - - - - - - - -
I need the answer fast please !
Answer:
30
Step-by-step explanation:
Angles on a straight line are equal to 180°
180°-120°= 60°
Sum of Co interior angles are equal to 180°
180°-60° =120°
90°+60°+ x = 180°
((180° - 90° - 60°= 30°))
A company that makes hard candy have a standard bag of hard candy with 150 pieces. The hard candy has three distinct colors red, white and orange and equal proportion of each candy is present in the standard bag. The manager wants to know whether the bags produced last Monday were similar to the standard bag. To test this, they plan to choose a random bag from the batch and compare it with the standard bag. You will have to answer questions below to assist the manager in comparing the two bags. Based on the data provided above how many degrees of freedom will you have while performing the necessary test of comparison
Answer:
The degrees of freedom, Df = The number of bags produced on Monday - 1
Step-by-step explanation:
The number of degrees of freedom is the limiting number of values that are logically not influenced by other values such that they are capable of having variation
The degrees of freedom = The sample size - 1 = N - 1
Therefore, the degrees of freedom, Df = The number of bags produced on Monday - 1
A bag contains 12 yellow tiles and 12 blue tiles. A student will choose one tile from the bag without looking. Which word(s) describe the probability of choosing a blue tile from the bag? likely O certain O impossible O equally likely
on a k12 quiz btw
Answer:
Equally Likely
Help please please pray that you pray
Answer:
well religion does not work here sorry
Step-by-step explanation:not everyone prays im sorry
Answer:
im confused
Step-by-step explanation:
dont delete this
Ben needs to replace two sides of his fence. One side is meters long, and the other is meters long. How much fence does Ben need to buy?
Answer:
696 39/100 meters
Step-by-step explanation:
Ben needs to replace two sides of his fence. One side is 367 9/100 meters long, and the other is 329 3/10 meters long. How much fence does Ben need to buy?
Side A = 367 9/100 meters
Side B = 329 3/10 meters
How much fence does Ben need to buy?
Total fence Ben needs to buy = Side A + side B
= 367 9/100 + 329 3/10
= 36709/100 + 3293/10
= (36709+32930) / 100
= 69639/100
= 696 39/100 meters
Ben needs to buy 696 39/100 meters
A researcher wishes to estimate, with 90 % confidence, the population proportion of adults who eat fast food four to six times per week. Her estimate must be accurate within 2% of the population proportion. Find the minimum sample size needed.
The minimum sample size needed is 423.
To find the minimum sample size needed to estimate the population proportion with a given level of confidence and a desired margin of error, we can use the formula:
n = (Z^2 * p * q) / E^2
where:
n is the minimum sample size
Z is the Z-score corresponding to the desired confidence level
p is the estimated proportion of the population
q is 1 - p (complement of the estimated proportion)
E is the desired margin of error
In this case, the researcher wants to estimate the population proportion of adults who eat fast food four to six times per week with a 90% confidence level and an accuracy within 2% (margin of error of 0.02).
Since the estimated proportion is not given, we can use a conservative estimate of p = 0.5, which maximizes the sample size. This is because when the estimated proportion is unknown, assuming p = 0.5 results in the largest sample size required.
The Z-score corresponding to a 90% confidence level is approximately 1.645.
Plugging the values into the formula:
n = (1.645^2 * 0.5 * 0.5) / 0.02^2
n ≈ 422.94
Rounding up to the nearest whole number, the minimum sample size needed is 423.
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3. The following sequence shows the number of pushups Kendall did each week, starting with her first week of exercising: 6, 18, 54, 162…
(a) What is the recursive rule for the sequence?
(b) What is the iterative rule for the sequence?
{{{ THE BOLDED CHARACTERS SHOULD BE SMALL. }}}
SEQUENCE: 6, 18, 54, 162
18/6 = 3
54/18 = 3
162/54 = 3
then, r (common ratio) = 3
_________________________________________
RECURSIVE RULE: r = 3
an = a(n - 1) × r [formula]
ANSWER: an = a(n - 1) × 3
_________________________________________
ITERIATIVE RULE: r = 3, a1 = 6
an = a1 × r^(n - 1) [formula] [ ^(n-1) is an exponent]
ANSWER: an = 6 × 3^(n - 1)
Use the Divergence Theorem to compute the net outward flux of the vector field F = (x², - y², z²) across the boundary of the region D, where D is the region in the first octant between the planes z = 9 - x - y and z = 6 - x - y.
To apply the Divergence Theorem, we need to first find the divergence of the vector field F:
div(F) = ∂/∂x(x²) + ∂/∂y(-y²) + ∂/∂z(z²)
= 2x - 2y + 2z
Next, we find the bounds for the region D by setting the two plane equations equal to each other and solving for z:
9 - x - y = 6 - x - y
z = 3
So the region D is bounded below by the xy-plane, above by the plane z = 3, and by the coordinate planes x = 0, y = 0, and z = 0. Therefore, we can set up the integral using the Divergence Theorem as follows:
∫∫F · dS = ∭div(F) dV
= ∭(2x - 2y + 2z) dV
= ∫₀³ ∫₀^(3-z) ∫₀^(3-x-y) (2x - 2y + 2z) dz dy dx
We can simplify this integral using the limits of integration to get:
∫∫F · dS = ∫₀³ ∫₀^(3-x) ∫₀^(3-x-y) (2x - 2y + 2z) dz dy dx
= ∫₀³ ∫₀^(3-x) [(2x - 2y)(3-x-y) + (2/3)(3-x-y)³] dy dx
= ∫₀³ [∫₀^(3-x) (2x - 2y)(3-x-y) dy + ∫₀^(3-x) (2/3)(3-x-y)³ dy] dx
Evaluating the two inner integrals, we get:
∫₀^(3-x) (2x - 2y)(3-x-y) dy = -x²(3-x) + (3/2)x(3-x)²
∫₀^(3-x) (2/3)(3-x-y)³ dy = (2/27)(3-x)⁴
Substituting these back into the integral and evaluating, we get:
∫∫F · dS = ∫₀³ [-x²(3-x) + (3/2)x(3-x)² + (2/27)(3-x)⁴] dx
= 9/5
Therefore, the net outward flux of the vector field F across the boundary of the region D is 9/5.
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Find the amount necessary to fund the given withdrawals.
Semiannual withdrawals of 270$ for 6 years, interest rate is 6.2% compounded semiannually
Answer:
the amount that necessary to fund is $2,671.61
Step-by-step explanation:
The computation of the amount is shown below;
Given that
PMT = $270
NPER = 6 × 2 = 12
RATE = 6.2% ÷ 2 = 3.1%
FV = $0
The formula is shown below:
=-PV(RATE;NPER;PMT;FV;TYPE)
After applying the above formula the present value is $2,671.61
hence, the amount that necessary to fund is $2,671.61
Find the value of x so that the ratios 8 : x and 12 : 18 are equivalent.
Answer:
12
Step-by-step explanation:
12:18=2:3 multiply by four to get 8 as the first term
8:12
8:x
so x is 12.
*this could be wrong so check my work
Answer:
x = 12
Explanation:
8 : x
12 : 18
to find the value of x, we need to convert 12 into 8. since they both have a common factor of 4, we can divide 12 by 3 to get 4, and then multiply it by 2 to get 8. in numerical terms, this is:
12 ÷ 3 = 4
4 × 2 = 8
now, we can do the same for 18.
18 ÷ 3 = 6
6 × 2 = 12
thus, 12 : 18 is equivalent to 8 : 12. x = 12.
Approximate the area of the region bounded by the graph of f(t) = cos (t/2 - 3 pi /8) and the t - axis on [3 pi /8, 11 pi /8] with n = 4 subintervals. Use the midpoint of each subinterval to determine the height of each rectangle (see figure). The approximate area of the region is . (Round to two decimal places as needed.)
To approximate the area of the region bounded by the graph of f(t) = cos(t/2 - 3π/8) and the t-axis on the interval [3π/8, 11π/8] using 4 subintervals, we can use the midpoint rule.
The width of each subinterval is (11π/8 - 3π/8) / 4 = π/2.
We can calculate the height of each rectangle by evaluating the function at the midpoint of each subinterval.
The midpoints of the subintervals are:
t₁ = 3π/8 + π/4 = 5π/8
t₂ = 5π/8 + π/4 = 7π/8
t₃ = 7π/8 + π/4 = 9π/8
t₄ = 9π/8 + π/4 = 11π/8
Calculating the corresponding heights:
f(t₁) = cos(5π/16 - 3π/8)
f(t₂) = cos(7π/16 - 3π/8)
f(t₃) = cos(9π/16 - 3π/8)
f(t₄) = cos(11π/16 - 3π/8)
Now we can calculate the area of each rectangle by multiplying the width by the height.
Area of rectangle 1: (π/2) * f(t₁)
Area of rectangle 2: (π/2) * f(t₂)
Area of rectangle 3: (π/2) * f(t₃)
Area of rectangle 4: (π/2) * f(t₄)
Finally, we can approximate the total area by summing up the areas of all the rectangles:
Approximate area = Area of rectangle 1 + Area of rectangle 2 + Area of rectangle 3 + Area of rectangle 4
Please note that I cannot provide the exact numerical values as the calculation involves trigonometric functions and the specific values of π/8.
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You have an annual salary of $47,334. Your monthly expenses include a $1,115 mortgage payment, a $336 car lease payment, $112 in minimum credit card payments, and a $108 payment on your student loan. Calculate your DTI (debt-to-income) ratio as a PERCENTAGE (no % symbol needed).
Answer:
DTI ratio in percentage = 42.36
Step-by-step explanation:
Annual salary = $47334
This means that gross monthly pay = 47334/12 = $3944.5
Now,
Total monthly debt payments = 1115 + 336 + 112 + 108 = $1671
Debt to income ratio = (1671/3944.5) × 100% = 42.36%
Since we are told not to put the % symbol, then answer is 42.36