Tanya Has A Triangular Garden Bed In Her Backyard ✓ Solved
Tanya has a triangular garden bed in her back yard.
Tanya has a triangular garden bed in her back yard. The measure of the largest angle is 30° less than twice the measure of the smallest, and the measure of the middle angle is 10° more than the measure of the smallest angle. Find the measures of the three angles. Start your work by defining the unknown quantities in terms of a variable. Show all work and write a complete answer, including units.
Jacob decided to ride his bicycle across the country during his 3-month summer vacation. The route he took from Washington, DC to Portland, Oregon, covered a total of 3420 miles. To keep his mind occupied during some of the long flat stretches of countryside, he often did algebra problems in his head. One day, for example, he determined that his distance from his starting point was exactly 60 miles more than twice the distance remaining until the finishing point. How far was Jacob from the finishing point of his journey when he made that calculation? Start your work by defining the unknown quantities in terms of a variable. Show all work and write a complete answer, including units.
Melissa is planning to invest a total of $17,000 in two accounts. If she invests $10,000 in a CD paying 12% annual simple interest, at what rate does the remainder of her money need to be invested so that the two investments together yield at least $1900 in yearly simple interest? (Set up and solve an inequality to answer this question).
Paper For Above Instructions
Problem 1: Triangle Angles
Let the measure of the smallest angle be represented by the variable x. According to the problem, the largest angle can be expressed as 2x - 30° and the middle angle as x + 10°. The sum of the angles in a triangle is always 180°. Therefore, we can write the equation:
x + (x + 10) + (2x - 30) = 180
Simplifying this equation gives:
4x - 20 = 180
4x = 200
x = 50°
Now we can find the measures of the three angles:
- Smallest angle: x = 50°
- Middle angle: x + 10 = 60°
- Largest angle: 2x - 30 = 70°
Thus, the measures of the three angles are 50°, 60°, and 70°.
Problem 2: Jacob’s Bicycle Journey
Let y be the distance from Jacob to the finishing point. According to the information given, the distance he has traveled is described as y + 60 = 2(y - 3420). We can simplify this equation:
y + 60 = 2y - 6840
60 + 6840 = 2y - y
y = 6900
However, since Jacob’s journey was only 3420 miles, we need to revise the understanding of discrepancies. This equation leads to a situation where y is 60 miles more than twice the remaining distance until the finishing point. Therefore, we have:
y + 60 = 2(3420 - y)
y + 60 = 6840 - 2y
3y = 6780
y = 2260 miles.
Thus, Jacob was 2260 miles away from the finishing point when he made that calculation.
Problem 3: Investment Calculation
Let r be the rate at which Melissa needs to invest the remaining $7,000. The interest from the CD investment can be calculated as:
I = P r t = $10,000 0.12 1 = $1200
The interest required from the second account to meet the total interest goal is:
1900 - 1200 = $700
The interest equation for this investment becomes:
7000 * r = 700
r = 100 / 7000
r = 0.1 or 10%
Thus, the remaining $7,000 needs to be invested at a rate of 10%.
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